What is the difference between Microwave, Microwave Oven, and Oven?

First off, the terminology issue.

So, can someone please clear up this fog? What's the difference between Microwave and Oven and Microwave Oven?

"Microwave" is just short for "microwave oven". Both terms mean the same thing: an appliance that uses microwave radiation to heat food. Cooking food in this way is called "microwaving". An oven, on the other hand, has a heating element which heats the air inside, which then heats the food. Cooking food in this way is generally called "baking", though there are a lot of other things (e.g. roasting) you can also reasonably do in an oven.

So if all you have is a microwave, then all you can do is microwave (not bake). And if you have an oven, then you can bake (but of course you can't microwave).

There also exist combinations, which are capable of both microwaving and baking, i.e. they have a microwave emitter and a heating element. However, remember that in the US - and thus on much of the internet - something talking about a "microwave oven" likely does not mean this combination, but simply microwave heating. Names like "convection microwave oven with grill", "combination microwave/oven" do refer to these combinations, though. I've also seen "speed oven". You may sometimes have seen people referring to these combination microwave/ovens as a "microwave oven" (I think this may be common in Indian English), but that's really just imprecise language.

Some sites say that a "microwave" is only for heating or re-heating cooked food. Whereas, in "microwave ovens", you can microwave and bake. Is that true?

It sounds like in this statement, "microwave oven" is careless language, referring to a combination microwave and oven. (It doesn't make any sense at all otherwise; the two things are the same thing, so of course you can do the same things in them.)

So the question is really, what can you do in a microwave vs an oven, or a combination microwave/oven?

It's true that microwaves are mainly used for reheating cooked food, and it's true that they can't actually bake. However, they can do a lot more than reheat food; there are a lot of kinds of cooking besides baking. For example, microwaves are great at simmering/boiling and steaming, and it doesn't matter if the food was cooked already or not. The recipes you're talking about are probably mostly in those kinds of categories, but if you're interested, How do I know if a food or recipe can be made in a microwave oven? discusses in a bit more detail what works in microwaves.

The big things that you can do with ovens but not microwaves are the things that actually need the steady dry heat of baking. You can't bake bread or cookies or roast a chicken in a microwave. Microwaves hold in a lot of steam and don't get terribly hot, so you can't generally get things to brown or crisp up. They also don't really hold a temperature like you need for baking, they just pump more and more heat into the food.