That's no different from a normal vehicle.

: If it's to be nimble and fast, it won't carry loads of weaponry; it'll be
: smaller, lighter, less defended.

Yes, but what's it going to do that a Warthog can't?

: If it's to be a mule, it'll be strong, able to carry loads of weight, perhaps
: a lot of that being extra armour and protection; slow; depending on
: intended terrain, either not too flexible, or built to be multi-terrain
: capable.

Here, we seem to be talking about walkers, which I agree might have some use.

: If it's to be a tank, it'll likely be slower, heavier, built with stability
: and protection in mind, able to carry a decent amount of ammunition

Except if it's a tank analogue, there's still a number of factors working against it.
-You've got a taller profile. While the cross-sectional area presented to the enemy might not be much larger than a Scorpion, you can be spotted from further away, and are probably more noticeable in general.
-For a tank and a mech of the same weight/volume, the mech has more surface area, and will require less armor coverage or thinner armor to stay in the same weight range.
-Guns. A tank is well-designed for transferring recoil to the ground. Mechs can brace themselves if they have heavy cannons, but Scorpions don't have to. And yes, you could mount lower-recoil weapons like autocannons, missiles, mortars, etc, but what is a mech going to do with them that a conventional vehicle can't?

: etc etc. There's no single optimal mech. It's really just a class of very
: large machine housing a person and designed with a specific purpose.

Except that class comes with a number of design features that are, by nature, drawbacks. Those design features may offer benefits, but they're marginal at best.

A vehicle is being designed with a specific purpose. Alright. What design features promote mechs over vehicles for that purpose?

The sole instance (That this thread has actually convinced me of) where a mech would be more useful than a more conventional vehicle is in heavily forested woodland where the forest canopy would impede close air support.

: or intended to be an additional
: augmentation suit that provides even more capability to the Spartan over
: and above what the Mjolnir inherently provides.

I'll pass.

: I could accept either of those two latter contexts in a Halo video game :)

So, your criteria are:

Mech that is not designed for a Spartan.
Mech that is designed for a Spartan.

Anything more specific?