I used to work for Batteries Plus. Here's some info I hope helps:
Higher output is 100% dependent upon the specifications of the cells, not the chemistry. Regardless of voltage, maximum amperage output can differ between batteries. You can get two batteries that are the same chemistry, voltage, and manufacturer that will perform VERY differently because they have different amperage outputs. Manufacturers measure maximum output in C-rates, but some retailers will advertise an amp ("A") output rating instead of C-rate (a lot of times these numbers aren't very accurate). Usually batteries with high C-rates are called something like "high performance," "high drain," or "high output" cells. These batteries don't offer any advantages to to anything but HIGH drain devices (like airsoft guns or RC cars). Also, the SIZE of a cell determines its potential for maximum output. A 2/3A size cell (used in "small" packs), will NEVER be able to output the same amount as a Sub-C cell (large pack). The smaller cells just don't have as many materials to produce the output.
So when you ask which chemistry will give a faster trigger response, the answer is: any of them, depending upon how high output the cells are. Each type has many different advantages and disadvantages. However, it is getting to be more common for Li-Po batteries to be used these days as high-output batteries because they have such high energy density; they output a lot of power and hold a considerable amount of capacity in very small-sized cells. It is easy to find them with high-outputs (sometimes higher than you'll EVER get an airsoft gun to use), but they're also expensive, and not very resilient (or stable for that matter) in comparison to NiMH and NiCD. Hope that helps.
-BATMAN