Yes, the H501w can run on any AA-size battery - alkaline, NiMh (Eneloop or others), primary lithium, and lithium ion - but for the latter, the protected 14500 batteries I had were a bit too long so I had to use unprotected ones, so have to be sure not to run them down too far. For me, the best battery for this light is an Eneloop AA.
There are a lot of details and technical aspects, but you can consider that a CR123 has roughly the same energy as a primary lithium AA (like Ultimate Lithium).
The important thing, if you plan to use the light very much, is to make sure it can take both a rechargeable, and a primary, so you have flexibility. So in an H50 or H501 you could use any AA - alkaline, NiMh, lithium primary, or lithium ion (and probably carbon-zinc too). An H31 could use a CR123A, or a lithium ion 16340 (aka RCR123).
If you are hunting/camping and run down/run out of batteries, which type of battery do you think that the only little store for 60 miles around will have?