we should target our arguments according to the circles we find ourselves in
That's true, but I would also suggest that some arguments just don't need to be had. For example:
rather than jumping into extremely serious surgery blindly
I suspect this might be hyperbolic, but even so, this is not a position any actual person holds. Nobody who is in favor of transgender rights thinks that people should just "jump into extremely serious surgeries blindly." Repeating this framing of the problem in this way plays into the hands of people who are intentionally misrepresenting what the actual argument is in order to create division.
If you want to target your arguments to your audience, you should steelman the arguments you're targeting, otherwise you're just confirming their beliefs.
Point taken, that was somewhat hyperbolic, but there are large groups of people who believe that anyone who asks for gender reassignment surgery should immediately get it, and doctors who try to slow things down are evil/misguided.
I'm trying to find the ones that I saw a few years ago, but now all the results show up news articles about states banning various trans related therapies. It could be I'm misremembering (this was about 5 years ago), but I believe that I found this quite a common opinion at the time.
I've never seen anyone argue for that position. I'm sure there are people who think that, but they must be a small minority. I'm willing to be convinced that I'm wrong, though.
Ok, it's possible I'm misremembering some conversations I had a while ago. There's a large number of people who are complaining that gender affirmation therapy is provided too easily without due diligence (random example https://www.google.com/amp/s/news.sky.com/story/amp/woman-says-rushed-gender-reassignment-treatment-left-her-suicidal-11946763), but I seem to remember seeing a lot of articles saying the opposite, and discussing them with people - along the lines of "if somebody says they're trans they're trans, and we don't need doctors to decide whether they can get therapy".
I have also been called many things, often mutually contradictory. How I see it is that labels are relative. It is not me inconsistently "being X" in one community, and "being anti-X" in another community, but rather me consistently being me, which happens to be more X than the community #1, and less X than the community #2.
Like, applying to the wokeness debate or similar, I think that weird people should be left alone, but people who comment that something seems weird, unless they imply some hostile action, should also be left alone. That is quite consistent from my perspective: all people should be left alone, unless they try to hurt someone else. Sometimes it is the normies trying to hurt others, sometimes it is the weirdness warriors. But I can imagine two different groups of people yelling at me for being something, and the opposite of it, respectively.
Or, in context of religion, I don't believe in anything supernatural, but I also try to understand the specific statements of specific religions, so that I can tell people they are strawmanning X, without actually believing X myself. I can also explain the motte of some belief, while aware that they are playing the motte-and-bailey game.
Or I can support a cause, while distinguishing between things that are important for the cause, things that are less important, and things that are perhaps completely useless but for some reason are used as virtue signals. That again is not very popular, because for many people, virtue signalling is exactly why they play the game.
My opinions are often of the form: "I believe that X is often the right choice, but if you happen to be in the situation where X is the wrong choice, then obviously don't do X". Not popular. The popular positions are "X is always right" and "X always sucks". I am in favor of actually looking at the details, which again could be a popular position among people who call themselves wise and say "for any X, X is exactly 50% right and 50% wrong", but I can disagree with that too, and say that X is a good default, because it is right in 90% of situations.
Related: Lonely Dissent
Lonely dissent doesn’t feel like going to school dressed in black. It feels like going to school wearing a clown suit. That’s the difference between joining the rebellion and leaving the pack.
Except, the position is not necessarily lonely in the sense "only you think it", but can be relatively lonely in the sense "many people think it, but they do not feel the urge to make a Twitter army and persecute their opponents, so if you find yourself in a Twitter war it is often you alone against the entire army".
There seems to be a bit of divergence between your title ("locally woke, globally anti-woke") and the rest of the post (locally woke in some locales, locally anti-woke in some others, really nothing about "globally" at all).
[EDIT to note that the title has now been changed.]
From the title I was expecting something quite different (e.g., an argument that "woke" norms are best when interacting directly with others, but "anti-woke" norms make for better public policy). That's not to say that there's anything wrong with the actual post -- but I think it could use either a different title, or some content that connects more with the title. E.g., maybe the point is that the communities you're physically part of -- geographically local -- tend to be more conservative ones in which you look "woke", while the purely-online ones tend to be more progressive ones in which you look "anti-woke"; though that would feel more like "woke offline, anti-woke online" or something, and in any case seems more specific to the details of your personal circumstances than is necessary, since the general phenomenon you're talking about can apply to pretty much anyone.
[EDITED to add:] Though it might well be true that most online communities are quite progressive and most small local communities are quite conservative, in which case the specific situation you describe might generalize fairly widely. That seems anecdotally plausible but I'd want to see more actual evidence before actually believing it with any confidence.
Could there be a signalling component? Nobody you see online would ever be in favour of conversion therapy, so there's no risk for you to be mistaken for one of them. The ideology where one excludes anyone who doesn't support gay rights become the baseline, the least sophisticated ideology, so it's tempting to be a meta-contrarian and argue against it, which signals intelligence and freedom of mind. But IRL, you see that there are pretty homophobic people around (could be some family members at the Christmas dinner), so being a meta-contrarian is no longer an option, as it would just signal intolerance.
I notice a similar phenomenon, and I attribute it in part to the way that in-groups tend to judge their out-groups based on those out-groups' loudest, most controversial, or otherwise most newsworthy members. I find that the conversation I have with anyone about beliefs which differ from theirs tends to start with "so why is it that you think that all x believe y? how many x have you actually talked to about this?"
Unfortunately, the tendency of information to spread like this seems to emerge not from any individual's unique and addressable weaknesses, but rather from the systemic bias to pay attention to things which seem unusually extreme. It's like how what's shown on TV is selected foremost for making good TV, rather than foremost for truth or accuracy or any loftier virtue. Certainly each audience has a plausibility or evidence threshhold where they'll feel better about engaging with content if it Has Citations(TM), but the effort it would actually take to read all of the cited evidence for every argument they see is absolutely inaccessible to most of the population. Sure, maybe people technically "could" if they just Tried Harder And Were Morally Superior, or if The Education System hadn't Failed Them, or however you want to frame it, but ultimately they don't.
Like most people , I live in many different spheres which only barely interact. My community is very different from my workplace, is very different from the people I hang out with on the internet, is very different from my family.
Whilst my viewpoints don't change depending on who I'm interacting with, the direction my viewpoints point in often flips sign depending on what the other side believes.
For example I was bought up in an ultra orthodox Jewish community, and now live in a more modern, but still mostly orthodox Jewish community.
Growing up I had to fight hard just to convince my friends and family that gay people aren't evil, deserve to be able to publicly admit they're gay, and that conversion therapy is probably a bad idea. I now live in a community where that's not necessary, but I still often have to argue that you should respect people's chosen pronouns as a matter of basic respect, even if you might personally disagree with transgender politics.
Meanwhile when interact with people online I often find myself arguing that it's still ok to be friends with someone even if they personally don't support gay rights, and that it's worth considering the costs and benefits of gender reassignment surgery on an individual basis, rather than jumping into extremely serious surgery blindly.
From the perspective of my friends growing up I'm a bleeding heart liberal. From the perspective of some of my more extreme friends online I'm a cold hearted bigot.
This also applies to many other controversies. In some circles I need to persuade people that "no we shouldn't litter the environment", in others, that replacing plastic items with paper alternatives doesn't actually do much, if anything, to help the environment, and imposes quite a large cost in the process.
I don't really know if I have any major point here, other than suggesting that all debates are bravery debates and that we should target our arguments according to the circles we find ourselves in, rather than spreading the same message to everyone we meet.