OK, this is the UK perspective 'cos I don't have much experience of what happens elsewhere.
DrSyn has given the book definitions. As far as I know PAN MEDICO is not aviation phraseology - that is, ICAO doesn't recognise it - so it would be unwise to start using it in just a few odd countries. You could campaign for it to be adopted by ICAO but really it doesn't offer very much extra information. A PAN call says 'I've got a fairly urgent problem' - and ATC will want to know a bit more so that they can best deal with you. A PAN MEDICO call would say 'I've got a fairly urgent problem because someone's not well' - and ATC would want to know a bit more so that they can best deal with you.
In practice HD is spot on - although a PAN call is usually going to be the textbook answer, the important thing is to tell ATC what the problem is. If you say you've got a medical emergency, whether you prefix it with PAN or not, ATC will give you priority handling - whatever you say you want, rapid descent, diversion, gentle turns, slow descent or whatever, you'll get it as soon as it's possible. In almost every case ATC will call an ambulance (or notify someone else who will), but there's no guarantee - it all depends on what the Aerodrome Emergency Orders say. If you can't get through to your handling agent or you want to be sure an ambulance has been called, ask.
If you're operating somewhere where English is not the only language used on the RT then yes, making a PAN call is both textbook and going to get attention. A MAYDAY call is probably not the correct answer because the standard respose is for ATC to impose RT silence on everyone else which is not going to be necessary in this circumstance.