Further, the assumption is that he just cast Avada Kedavra against Lily and James. So that would have been the last spell if he cast no spells at Harry and just left.

If we're assuming a clever Voldemort, there's any number of ways that 'check his wand's last recorded spell' could fail to be conclusive evidence. He could've used someone else's wand (a backup wand is a good idea for a ton of reasons). He could've brought someone else to do the actual deed (how else did Bellatrix get to the wand before anyone else showed up like Dumbledore or Aurors?). He could've used any of a billion Muggle methods, as you point out (plausible since we know about Pioneer and Dumbledore regards clever use of Muggle tech as pointing to Voldemort). He could've used an innocuous spell in a fatal way (a dark lord in hiding could be expected to use Apparating all the time, and Apparating seems like it could be quite fatal given 'splinching'). He could simply have broken the detection spell and set up a false audit trail, as it were. And so on and so forth.