I've only seen the one with the bald doctor, but he just says he's the Doctor even though I'm fairly sure he didn't do any healing at that time. Why is he called a Doctor? Is he even a doctor in any way? Medically speaking and otherwise?
It's essentially a hen/egg problem:
Especially during the eleventh Doctor's time it's repeatedly implied (or at least hinted at), that the Doctor made a name of himself in the whole universe, literally branding his "name" on the profession of being a doctor, a man with great knowledge or a wise man helping and protecting others from their issues and problems.
To quote River Song from A Good Man Goes to War (Series 6; emphasis mine):
This was exactly you. All this. All of it. You make them so afraid. When you began, all those years ago, sailing off to see the universe, did you ever think you'd become this? The man who can turn an army around at the mention of his name. Doctor. The word for healer and wise man throughout the universe. We get that word from you, you know. But if you carry on the way you are, what might that word come to mean? To the people of the Gamma Forests, the word Doctor means mighty warrior. How far you've come. And now they've taken a child, the child of your best friends, and they're going to turn her into a weapon just to bring you down. And all this, my love, in fear of you.
The BBC Website covers this very thoroughly;
The Doctor has made many conflicting statements about his qualifications. He certainly studied human medicine, and in The Moonbase he says that he gained a doctorate under Joseph Lister in Glasgow in 1888. However, Lister left Glasgow years before that date, so the Doctor is either being vague or is lying.
He can carry out such basic tasks as resetting dislocated fingers (The Smugglers), but perhaps the archaic nature of his knowledge (he gives incorrect advice on fever treatment in The Ark and can't so much for blood poisoning in The Myth Makers) leads him to deny being a doctor of medicine (An Unearthly Child, The Krotons). His advanced medical efforts in The Sensorites rely largely on chemistry skills, and he has local medics to help him.
The other possibility is that he failed to get an attempted medical degree (as stated in The Rescue) and, knowing the condition he was treating in The Moonbase, exaggerated his achievements. In Robot, knowing he has a doctor on board, he renounces all claims to a degree save a 'purely honorary' one, which turns out to have been acquired from St Cedds, Cambridge, in 1960 (Shada).
However, at some point the Doctor does gain the advanced medical knowledge he uses in The Twin Dilemma and The Trial of a Time Lord. Perhaps the gap between The Deadly Assassin and The Face of Evil (see The Doctor's Age) is the most likely place for this to happen, as he's uncertain about a broken arm in The Seeds of Doom, but familiar with 50th century medicine by The Invisible Enemy. By the time of Remembrance of the Daleks he's learnt to diagnose by touching someone's ear.
Before leaving Gallifrey, the Doctor gained a doctorate of some sort from the Prydonian Academy (The Hand of Fear, The Deadly Assassin), qualifying, with Drax, in the class of '92 (The Armageddon Factor). His teachers included Borusa and Azmael (The Twin Dilemma), the latter possibly being the tutor who understood artron energy (Four to Doomsday). Since he knows so much about law (The Deadly Assassin, The Stones of Blood) it is possible that he qualified in this area.
However, on many occasions the Doctor has called himself a scientist (Planet of Evil) and engineer (The Aztecs, The Mind of Evil). Perhaps he's telling the truth about a combined degree when he calls himself a doctor of many things (Revenge of the Cybermen).
In the modern series, the Doctor is more vague, simply referring to himself as a "Doctor of everything", specialising in "making people better".
I think the show's original creators meant "The Doctor" in the academic sense of having an advanced degree, i.e. a doctorate (see http://en.wikipedia.org/wiki/Doctor_(title) ), based on the character's expertise in many fields, especially scientific ones. As mentioned on p. 5 of the book Doctor Who: A History, Sydney Newman was the one who first came up with the idea of the character, and his original conception was that the character was an alien scientist who couldn't remember his own name, who just called himself "The Doctor". In the show's very first story "An Unearthly Child" there was this exchange:
IAN: All right, now we're helping them. You're a doctor, do something.
DOCTOR: I'm not a doctor of medicine.
The article "The Doctor" from the Doctor Who wiki talks about what The Doctor may have had doctorates in:
The title "Doctor" was not undeserved; he did hold one or more doctorates of some sort, (TV: The Armageddon Factor, The God Complex) formally studied medicine on at least 19th century Earth, (TV: The Moonbase) and frequently displayed detailed medical knowledge. (TV: The Ark, Frontios, The Empty Child, New Earth, The Time of Angels, The Curse of the Black Spot) At least some versions of his sonic screwdriver performed medical scans and healed minor wounds. (TV: The Empty Child, The Vampires of Venice, A Good Man Goes to War) He showed knowledge on how to help someone thrown by an explosion recover quickly. (TV: Remembrance of the Daleks) Although his first, (TV: "The Forest of Fear", "Mighty Kublai Khan"), fourth (TV: The Ark in Space) and fifth incarnations (AUDIO: Red Dawn) had claimed not to be a doctor of medicine, and his third (TV: Spearhead of Space) and tenth incarnations (TV: Utopia) claimed to be a doctor of practically "everything", by his eleventh life the Doctor claimed to hold doctorates in at least medicine and cheese-making. (TV: The God Complex)
And note that The Master (one of his major long-term rivals who you may not have seen if you've only watched the Eccleston series) also has a traditional academic title, albeit a less advanced one (edit: but see armb's comment below, the title could also be a reference to the Master of a College rather than just someone with a Master's degree).
In the Spearhead from Space, when Liz Shaw asks the Third Doctor what he is a doctor of, he replies, "Practically everything." This might imply an inclusion of medicine, but that would contradict the First Doctor's statement in An Unearthly Child that he is not a medical doctor.
Furthermore in The Ark in Space, the Fourth Doctor remarks that his degree is honorary.
In spite of any medical knowledge the Doctor has, it's uncertain whether or not he has an actual MD. Likely he is not a "real" doctor in the MD sense, but this is subjective given the nature of time travel (the common anatomical knowledge of a 21st Century American might exceed the medical knowledge of a 17th Century physician).
Scientifically speaking, he consistently demonstrates tremendous aptitude. Again, the qualifications for a doctorate are subjective depending on era and planet. In The City of Death, Dr. Kerensky speaks of himself as being the foremost expert on temporal science on Earth of 1979, but his knowledge with a doctorate clearly pales compared to the Doctor's.