I think having 2 different reputation listings would just complicate things, and lead to all kinds of questions of how to combine or test them for reputation based rewards. I however agree with the basic premise going around that the "ask questions only" reputation gain poses a problem:
People attach a value to reputation. When they see people gaining it without "earning" it, it makes them feel that their own hard earned reputation has been cheapened. This kind of negative resentment is not good for a healthy community.
People want to gain reputation. Right now the easiest way of gaining reputation without the need for skill or effort is to post a lot of mildly controversial or just "what is your personal opinion of" conversational questions. While these kinds of questions are just within the rules, they're not what the system is for - but unfortunately like gold farming, the rules and rewards of the system encourage this undesired behavior from those who don't feel like playing fairly.
Most importantly reputation controls access to various moderation features. Since there is yet to be a solution to reputation inflation, anyone who just sits around doing nothing but asking off-base questions will have moderation control over other users (and sooner rather then later based on some of these users). This I think is the breaking point, as these types of users have not shown they are a productive member of the community (they've done the opposite), and yet have moderation control over others. See also "negative resentment" above.
I would propose a different type of solution, one that preserves the reward for asking good questions (which is a needed component for the community), but prevents that mechanic from being farmed - use a reputation cap just like the one used to prevent a single answer/question from overly inflating a persons reputation.
The rate cap would need 3 components:
A per user counter for how many reputation points the user has earned from questions. This would be the same as the counter used to check the 200/day cap, but it would only count questions and would not get reset each day. There would be the issue of calculating this value for users when the system first went into use.
A constant value for a question grace amount, say 499 or 999.
A constant value for a question/answer ratio, say 50%.
Combined with the users actual reputation these values are used to calculate a floating rate cap for question upvotes:
Question Rate Cap = Total Reputation * Question/Answer Ratio + Question Grace Amount
This rate cap is then applied to question upvotes in the same manner as the 200/day rate cap (by comparing the cap to their counter before applying the upvote). Users are free to gain reputation from questions, but they cannot gain moderator powers (or otherwise appear as high reputation users) until they start adding some reputation from other sources, at which point they can resume gaining reputation from questions as well.