Alkyds: These are paints made with a synthetic resin instead of a natural drying oil. The chief advantage to alkyds is that they dry overnight, and all colors dry at the same rate. The big disadvantage is that (to me) they smell awful. The handling is also inferior to the handling of high quality oil paint. Paintings done in alkyds should be labeled as alkyd paintings, not oil paintings.
There are also alkyd mediums, such as Liquin, Galkyd, and Neo-Meglip, intended to be mixed with regular oil paints. That's not what I'm talking about here.
Water miscible oil paints: These are oil paints made with a form of linseed oil that has been modified so that, when water is mixed in, it doesn't separate. It is therefore possible to clean brushes in water rather than solvents, clean your palette with water, and so on. It is also possible to thin the paint with water, although manufacturers usually recommend against adding a whole lot of water. When mixed with water, the paint forms an emulsion (tiny droplets of water surrounded by oil), so the refraction index of the paint changes. That means that there is a noticeable shift in value; dark colors become a bit lighter. The paint returns to its normal value when the water evaporates away, making it difficult to judge values when painting (acrylic paint—another sort of emulsion—has the same problem). They take about as long to dry (oxidize) as regular oil paints. Water miscible oil paint can also be thinned with regular solvents, and manufacturers produce various mediums. They can even be mixed with regular oil paints, although no water should be added to such a mixture. Oil painters who try water miscible oils often find them to be kind of "sticky." Because each formulation is different, it can be a bad idea to mix paints from different brands. The big advantage to water miscible paint is that cleanup is easier and, well, cleanup is easier. Because the oil is real oil, I don't consider it unethical to label a painting made with water miscible oil paint as an "oil painting." So you don't have to try to educate buyers about a medium they’ve never heard of. Overall, though, I think water miscible paint is a solution in search of a problem.Many people on internet art forums mistakenly refer to water miscible oil paint as "water soluble" oil paint: that’s not technically correct, any more than there is such a thing as water soluble olive oil. Many people who use the terms “water soluble” and “water miscible” as if they mean the same thing misspell "soluble" as "soluable." I hate that in the depths of my pedantic little soul. Remember: “soluble” does not have an “a” in it.
Heat set “oil paints:” This is a line of paints marketed by one company: Genesis Artist Colors. I have not tried them. These are not actually oil paints, although the company describes them as “heat set artist oils” on their web site. The paints are instead ground in some sort of synthetic polymer that behaves rather like oil. It does not, however, dry by oxidation, the way oil paints do. In fact, it doesn't ever dry until you heat it to a high temperature, at which point it sets permanently. So you can leave paint on your brushes as long as you want. You don’t ever have to clean your palette—the paint stays wet forever. When you want a painting to be dry, you use a special heat gun (sold by the Genesis company) or a special drying oven (sold by the Genesis company). You can use a regular oven, but eventually, of course, you’re going to get paint on the inside of the oven. You can’t use a hair dryer because it doesn’t get hot enough. The big advantage of heat set paints is that you don’t have to worry about cleanup until you feel like it. When my son was born last week I left some oil paint on my palette: it’s now hard and will be a pain to scrape off. With heat set oils, that wouldn't be a problem. Some artists also like to noodle around with wet oil paint for days. Heat set paints don’t dry until you tell them to. The disadvantage to heat set paints is that they are made by only one company, and they won’t say exactly how they are made. They claim the paint is archival, but you have to take their word for it. Another disadvantage is that when you label one of these paintings, it would be a lie to say they are made with "oil paint." I'm not sure what you should call them—"heat set paint," maybe. I think its unethical for the company to call them “heat set artist oils,” because they are not oil paints, however much the finished product may look like an oil painting.
All of these paints are marketed to hobbyists, who like the idea of oil painting, but want something less inconvenient. Each of them corrects some perceived flaw in oil paints: they take too long to dry, they dry when you don’t want them to, you have to clean up with smelly stuff, and so on. These kinds of “improvements” mostly appeal to hobbyists who want their hobby to be more convenient. That isn’t to say that there aren’t professional artists who use each of these types of paint, or that wonderful paintings aren’t made with them. But any company hoping to make a profit selling what I will call “convenience oils” has to market them to hobbyists. That means they have to be fairly inexpensive, so the quality of most of these paints is about equivalent to student-grade oil paint. In order to keep the price low enough that hobbyists will buy it, alternative pigments, cheaper grades of pigment, and extenders are used, just as with student grade oil paint. Certain pigments that are more toxic, such as lead white and genuine vermillion, aren't manufactured in convenience lines of oil paint for the same reasons you don't find such pigments in student grade paint: most hobbyist painters are afraid of toxic chemicals they don't understand. It would be possible for companies to manufacture convenience oils with very high standards of quality, but that would be a bad business decision. (I am told that the water miscible paint made by Holbein is of fairly high quality, but I prefer not to use a type of paint made by only one company).
For that reason, I won’t use them. I like having access to the best grades of artist-quality oil paint. I like being able to use a wide range of traditional mediums that I can mix myself. I like having access to traditional pigments that have valuable properties but require particular care with regard to safety. And I like being part of a tradition of painting that goes back to the early Renaissance. While real oil paints can be inconvenient, none of the alternatives that have been developed are, to me, worth giving up the real thing.
Update 1: I'd like to note that water miscible oil paint can be useful if you want to travel by air with a set of oil paints. Because they are easier to use without solvents (which they won't let you travel with) water miscibles can be an alternative that is relatively hassle-free. If I were to do that, I would definitely buy the Holbein Duo Aqua oils, which I have heard good things about.
Update 2: One other characteristic of alkyd paints that I forgot to mention is this: alkyds need more binder to a given amount of pigment than oil paints do. That means paint manufacturers can't use as much pigment when making alkyd paints, so some colors don't have the saturation and intensity of their oil equivalents. That's not a failure on the part of manufacturers, but a characteristic of the alkyd medium.