Like all language, and especially technical or domain-specific language, you need to know your audience/correspondent well enough to guess which terms are understood in the way you expect, and which ones need clarification. I fully support you, if A is worth the effort to educate. Many As are not.
For this topic, Wikipedia is a reasonable authority and makes it clear that ability to modify/distribute is a core part of open source as commonly used. Sadly, the Open Source Initiative never got a trademark on it, but has been fighting this fight since the previous milleneum.
Every time someone releases code publicly under some kind of "look but don't touch" terms a similar argument plays out:
A: This is cool, X is now open source!
B: It's cool that we can read it, but we can't redistribute etc so it's not "open source".
A: Come on, if it's not "closed source" it's "open source".
B: That's not how the term "open source" has historically been used. This is why we have terms like "source available".
A: It's bizarre that "open" would be the opposite of "closed" everywhere except this one term.
I'm generally with B: it's very useful that we have "open source" to mean a specific technical thing, and using it to mean something related gives a lot confusion about what is and is not allowed. While A is right that this is a bit confusing, it's also not unique to open vs closed source. Some other examples:
If a country doesn't have "closed borders" then many foreigners can visit if they follow certain rules around visas, purpose, and length of stay. If instead anyone can enter and live there with minimal restrictions we say it has "open borders".
If a journal isn't "closed access" it is free to read. If you additionally have specific permissions around redistribution and reuse then it's "open access".
If an organization doesn't practice "closed meetings" then outsiders can attend meetings to observe. If it additionally provides advance notice, allows public attendance without permission, and records or publishes minutes, then it has "open meetings."
If a club doesn't have "closed membership" then it's willing to consider applicants. If anyone can join who meets some criteria, it has "open membership".
This is just how language works: terms develop meanings that are not always ones you can derive simply from the meanings of their component words. I agree it can be confusing, but I also want to do my part to resist semantic drift and keep "open source" matching its useful and socially beneficial definition.