In my experience, jin4 is a bit more often.Both jin3 and jin4 can be heard in life, and I think jin3 is more often.
The confusion came from the simplification of Chinese characters. 儘jin3 and 盡jin4 were simplified to the same character 尽(jin3 and jin4).
This is why I think it's unnecessarily complicate and vague.For 【jǐn】, used as a adv. For example, 尽早，尽可能，尽管. Usually put it before adjective, means try to be possible/ to be in the range of some extent.
尽早：as early as possible,
尽可能: try to make it possible
For 【jìn】，used as a verb. For example, 尽力，尽心，尽职. Usually put it before noun, means to be in superlative degree.
尽力/尽心： try your best;
用尽：run out of ;
想尽办法： try all the methods ;
路的尽头: the end of the road
For 尽快， in my opinion, both pronunciation are available. When it's 【jìn】, tone is much stronger that 【jǐn】. We use 【jìn】commonly.
I agree that jin3 and jin4 are often exchangeable in life. But strictly speaking, the distinction does exist.In 汉典,
儘(jin3)'s meanings include: 1.极；2.力求达到最大限度。
盡(jin4)'s meaning include: 2. 到达极端；3.全部用出，竭力做到；4.都，全。
Aren't they all too similar? By this definition, can you tell whether it should be jin4量 or jin3量?
If we use your definition, it would be jin4量, but 汉典 says jin3. I really don't think this differentiation is necessary.