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.
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?