There's a reason why we don't have such a word: too much incentive to cheat. If you bother to reply to an argument, you generally want to communicate one of two things: a. I paid significant attention and should be taken seriously in my comments. b. I didn't pay much attention, so I do not want to be taken very seriously and do not want to be thought dumb if I make a mistake. A is much more common than B, because strategy C (keep your mouth shut) is usually better than B. There is little advantage to gradation. If you want to be taken seriously, you'd like to communicate that you've paid at least as much attention as everyone else. So there is strong incentive to cheat at characterizing the attention you've paid.
Thus, communication of that sort needs to be expensive (i.e. inefficient). Typically, it takes the form of detailed analysis.
On the one hand, there's incentive to cheat, and on the other, once you've made a definite claim about how much attention you've put in, it's easier (and tempting) to check on whether you've actually checked the math or whatever.
The word 'grok' doesn't only refer to arguments, but it is used that way to mean 'reached an intuitive understanding'.
I like that one, although I prefer the less parochial term "assimilate". Also, for people who are already aware of the concepts I've described in my ranking system, you could say whether you've reached level 2 on the issue aka have plugged it in to your existing models.
Don't forget a term for not understanding despite putting forth a painful effort: banging head on wall.
Worked out logical implications of, in terms of anticipated experience...
Did the above and compared that to actual experience...
Evaluated for coherence (e.g., made sure that the same word isn't being used to mean two different things in the same argument)...
Evaluated for internal soundness -- that is, confirmed that the premises really do imply the conclusion, even if it turns out the premises are false...
Identified known arguments with a "family resemblance" that may form a viable reference class for evaluation or comparison...
There seems to be a hole in English (I don't know about other languages) for efficiently describing how much attention you've paid to an argument.
I can think of a variety of amounts and sorts of attention-- looks cool and plausible, examined for attractive arguments against the other side, checked math for accuracy, checked math for plausibility of application, looked for counter-arguments, looked for evidence in favor, checked factual material, checked for whether some parts are sound even if there are some errors.....
Any other suggestions for types of attention?