Seems I've heard an occasional complete drain-down and recharge
will extend the life of a rechargeable battery, that being constantly
partially discharged and recharged is what shortens their life. Not
so? Does it depend what kind of rechargeable it is?

You're thinking of nickel-cadmium batteries. The lead-acid batteries used in
these power supplies can be crippled or destroyed by a "full" discharge. Try
to avoid it.