Here's my situation: In a fight with a group of vampire thralls, the party's wizard got caught in a corner and was being savaged by vampire bites, his max HP dropping from 24 to 11. They fended off the vampires, but the wizard was at 3hp (He refused to be healed by the cleric due to his character's hatred of religion and gods). He activated a trap collapsing the temple, and ended up getting hit by a falling chunk of stone ceiling, taking 15 damage (the rock rolled better than any of the vampires).

Now, the wizard is reduced to 0 hp, with 12 damage left over. The cleric's player says that exceeds the wizard's current max hp of 11, causing insta-death. The wizard's player argues that the death threshold for negative HP isn't affected by max-hp-reducing spells, claiming that would make those kinds of spells more powerful than intended.

I have stories planned in either case, but I'd rather be certain that I'm following the rules.

Is the threshold for instant death based on current max hp or normal max hp?

Yes, anything that lowers your max HP reduces the threshold for instant death

The rule on instant death from massive damage says:

Massive damage can kill you instantly. When damage reduces you to 0 hit points and there is damage remaining, you die if the remaining damage equals or exceeds your hit point maximum.

For example, a cleric with a maximum of 12 hit points currently has 6 hit points. If she takes 18 damage from an attack, she is reduced to 0 hit points, but 12 damage remains. Because the remaining damage equals her hit point maximum, the cleric dies.

The rule makes no differentiation between your current max HP and your regular max HP, and makes no exceptions for effects that reduce your max HP. (I also don't know of any max-HP-reducing effects that make an exception to the Instant Death rule either.)

At any one time, you only have one "hit point maximum", though certain effects may reduce it (and spells like greater restoration may end such effects). As a result, if your max HP is reduced, it does decrease the threshold for instant death from massive damage.

For instance, let's say your normal hit point maximum is 20, but a vampire bites you for 7 piercing and 10 necrotic damage. Its Bite attack reduces your max HP by the amount of the necrotic damage taken, so your new hit point maximum is 10. If you are damaged in a single instance that takes you to 0 HP and there is at least 10 damage leftover, you will indeed die instantly.

Yes

When damage reduces you to 0 Hit Points and there is damage remaining, you die if the remaining damage equals or exceeds your hit point maximum.

You only have one hit point maximum which is defined in the section on creating your character:

You start with hit points equal to the highest roll of that die, as indicated in your class description. (You also add your Constitution modifier, which you’ll determine in step 3.) This is also your hit point maximum.

Things that change your hit point maximum are acting on that number, the same one used in determining death from massive damage. Such changes include the additional hit points you get for gaining a level, certain spells and, yes, a Vampire Spawn’s Bite.