Do I have a branded smartphone (HTC Desire)? When my HTC Desire boots, I can first see a purple Android (?) icon, then a white background and the 'HTC' text, then a short animation with text 'T-Mobile'. Does that mean that I got a branded phone? The rest of the system doesn't show any signs of T-Mobile and I'm on a Vodafone prepaid 'contract'.

I'm worried that I got a branded phone (which means I'll get Froyo later than other HTC Desire owners) even though I'm not with T-Mobile.
Being that you get a T-Mobile graphic then I would say yes you have a branded Desire. Having said that my father has a branded O2 Desire and he has already received the Froyo update (today - Monday). I too have an O2 branded Desire, but I have not received Froyo yet which I suspect is because I am on a Business plan with O2, whereas my father has a personal plan.