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?
Yes :-)
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.