I want to drill holes in MDF of varying thicknesses, laid out to sub millimetre precision. In other words, I need the positioning of the center of the holes to be accurate to within a half millimetre (0.5mm) or better.
That's mostly a matter of accurate layout and a starter mark/hole (a single push from an awl can be sufficient) than a call for one particular style of bit.
So with regard to your question:
So would it be best to use brad point wood bits with centre punched holes as in woodwork, or HSS bits with pilot holes as in metalwork?
All bits that you could use, from basic twist bits through brad-point to auger can achieve the required accuracy.
I'd add both flat/spade bits and Forstner bits to that list as well, but only if you can use a drill stand or drill press. It's possible to control both with a power drill used freehand, but it requires experience and maybe a bit of luck and as such I would not recommend it where accuracy is paramount.
My problem is that I can't decide whether MDF is considered a wood or a grain free composite material.
It's closer to the second option here. It's not quite grain free, see previous Question Does MDF have a grain direction?, however it's close enough that it doesn't matter much in practice.