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.
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?