1995 Camry won't start

1995 Toyota Camry, 125,000 miles, 4 cyl, purchased used about 6 weeks ago. Previous owner says one day it was dead, he the had battery replaced and it was fine after, but his wife made him get rid of it anyway. Generally seems to have been well maintained. New battery 4 months ago but it seems to have an excess of powder-blue corrosion on the neg terminal of the battery.

Anyway, when I try to start it with the hood closed, I get nothing. With the hood open I get a buzzing noise. The "door open" light and alarm work with the key in, the radio works, but the headlights don't. When I try the headlights (no key) I get a buzzing in the headlight relay that builds up into a clicking noise. No dim lights- just nothing.

If it was the alternator, shouldn't I see a weak battery?

Checking the battery with a multimeter, it reads 12.65 Volts, so I didn't want to try jumping it or recharging the battery. I cleaned the corrosion off the battery terminals, but no change.

If it was the starter, shouldn't the headlights work fine?

What am I missing or are my assumptions just wrong?