Theres no hard rules, only etiquette. It depends on your tastes.
I have 4 cup types that I use:
- Duralex Picardie 220ml latte glasses - for lattes strangely enough
- ACF Cappuccino 190 ml - these are more bowl than tulip shaped so may be what you are after. Good and chunky and hold the heat.
- ACF Cappuccino 160ml (white and gold) - more tulip shaped
- ACF Espresso 70ml tulips for espressos
plus for everyday, I have an ex-Vinnies pair of Royal Doulton Africa series cups - about 200ml - that I find are good (especially at 50c each ;)). Once again thick and hold the heat well.
I have the odd double-walled glass ones (basically survivors) but I favour the Duralex or ceramics. I also have a set of 4 ceramic 220ml latte style cups (ex-Woolies) that I use for the occasional chai latte.
I wouldnt worry about using a special cup for flat whites. Use a latte glass or a ceramic cup.
Psychological or not, I find I enjoy my ceramics more than the glass.
FWIW, Ive bought most of my cups from Jetblack but other sponsors stock them.