In addition to using Mark Hubbard’s “Thank you in advance for [keeping me informed]” (or just “Thanks in advance for [ … ]” in less formal situations) to express your thanks in advance without having to rely on context, you could also consider using a near-synonym for being thankful to use with the conditional would, for example:
“I would (I’d)[ greatly/very much] appreciate your keeping me informed.”
or (to emphasize the conditional notion even further):
“I’d be [extremely/very] grateful if you would (you’d)/could keep me informed.”
(Please note that, although it’s probably overstating the actual level of appreciation/gratitude, I think I would include greatly and extremely, especially in the above, “in advance” cases, but probably less so in the “after the fact” cases mentioned below. However, be careful not to “over-do” it because you could come across as being sarcastic or “sycophantic” [the English sense!], so on second thought, maybe the more restrained very/very much would be better in these cases where intensifying the gratitude might be in order because you are actually requesting a future favor at the same time.)
Regarding “after the fact” expressions of thankfulness (such as the one you made on “Overflow” and are asking about here), in addition to Mark’s use of the past tense (provided) or relying solely on context (which I agree can be ambiguous, especially in your example where it could easily be interpreted as an on-going request), you could consider changing “keeping” to “having kept” and “keep” to “kept” to use either with the same near-synonyms above (but in the unconditional present) or else (preferably, I think) simply with the original “Thank you/Thanks” construction, for example:
“I [do] [greatly/very much] appreciate your having kept me informed.”
(where the do would be an optional way to emphasize the appreciation)
“I am (I’m) [extremely/very] grateful that you kept me informed.”
Thank you (Thanks) for having kept me informed.
(But please see this relevant Word Reference thread that seems to confirm that we are, in fact, often (but not always?) at the mercy of sometimes ambiguous context in English (as indicated in user3169's answer). At least “Kelly B” begins his/her answer with “Yes, that's right,” so making the effort in our own speech to avoid the ambiguity would be not only admirable, but also grammatically and logically correct, although we'd still have to rely on context to interpret the meaning when spoken to us by somebody who is less concerned with clarity.)
Best Answer
If I understand correctly, you are looking for a way to phrase the sentence that will follow the one you gave under context.
There are two issues that should be addressed with these phrases:
The potential mix up of various uses of the nouns way and path
which is not the meaning you are looking for;
In this sense it is used: The way to solve/resolve the issue. As in: There can be 2 ways to resolve the issue of energy shortage: finding an alternative source or reducing consumption.
You do not follow a way (this is the problem with your option #2). A way that you follow is usually a physical one (just follow the way until you reach the next crossroad, than turn left). (!There are exceptions to this, but they don't apply here.)
If there are several ways to do something, you can choose one. Although at first your option #1 sounded a bit ambiguous (the way we chose - is it referring to the way that you chose or the choosing strategy you applied?), in this context it is clear what you mean, so this option is fine.
This is often used to describe someones career path for example.It is more idiomatic to say that you follow a path than a way in this sense. Again, if there are several possible paths, you choose one of those.
You might say something like:
While this is a grammatically correct sentence, and there are examples of usage to follow a path in various texts (in social science, physics, law etc.) I am not sure this is the best (most natural sounding) way to put it. That is, it may not be a bad sentence, but it may not be a perfect one either.
Having considered all this...
Essentially it seams that your examples convey the same meaning as the sentence you provided in the part you marked as context:
The answer to this question = the approaches we will offer
the way we view the situation => the choice of the way/path to solve this problem
way/path = approach
So, unless there is a need to say this twice, the sentence: Depending on the way we view the situation, we will offer radically different approaches to this problem. is by far the best one.