That's very sad, mC. Many people whose treatment doesn't work as it is supposed to and is painful, like chemotherapy, tend to try alternative methods, because they seem to be a bargain. No side effects, pure healing.
Actually, the definition I did not give isn't vague. What is vague is the concept many people have of homoeopathy, which is not their fault.
The options shouldn't be limiting. I tried not to give too many, and thought I covered everything from "complete poo" to "only reasonable method" without putting fifty shades of grey between those extremes. I may have failed there. Which other options would you have liked?
The difference is, that there is proof that vaccines work, whereas there is no proof that homoeopathy does.
Or, in more detail: While vaccines contain non-infective pathogens, which the body's immune system then learns to recognize so that it can fend of the real beasties when they arrive, homoeopathic medicine contains (or rather, very much does not) some stuff (plant extract, mineral, secretion etc.) that, if taken undiluted, allegedly cause the symptoms of, say, a flu. So a homoeopathic flu cure would not actually attack the root of the flu (the virus). The reasoning is, that the homoeopathic drug, figuratively speaking, gives the body a kick to get it off its arse and start doing something against the symptoms. So, the drug isn't actually healing you, you're doing it yourself. The drug only activates the body's healing powers. Allegedly. Apart from that, vaccines are more isopathic than homoeopathic, because for a flu you get flu viruses.
This might sound logical to a point. On the other hand, would you take a poison that has the same effect as atropine (deadly nightshade) if you had eaten one of the berries?
Moreover, in a vaccine, you actually have a measurable number of pathogens, whereas in homoeopathic drugs, you often don't have anything. You might start with, as Hahnemann did, cinchona, which, consumed undiluted, allegedly causes the same symptoms as malaria. Then you dilute the hell out of it. The more you dilute it, the more powerful the drug becomes. There are dilutions based on powers of 10, 1:10, 1:100, 1:1000 (D1, D2, D3), on powers of 100 1:100, 1:10000, ... (C1, C2, ...) and on powers of 50000 - 1:50000, 1:2500000000 ... (LM I, LM II, ...). With dilutions higher than D24 (not uncommon) you statistically get less than one molecule per bottle - if you started with 100ml of, say, snake venom of a concentration of 1mol per litre, you'll get 1 molecule of snake poison in about every second of the 100ml D24 dilutions you made from that, none in the others.