The answer is none of the above. You can have warm air that is lower pressure, higher pressure, or equal pressure to cold air. However, if you want to focus on a single body of air (such as the air in a room, or air in a balloon), we can talk about how changes affect pressure.
The key equation, as I.E.P. mentioned in the comments is $PV=nRT$ where:
- P is the pressure of the gas
- V is the volume of the gas
- n is the number of mols of gas
- R is the ideal gas constant
- T is the temperature (in kelvin)
So we can see from this equation that if you take a fixed amount of gas (n doesn't change), and you increase its temperature by heating it, either pressure or volume (or both) must change. If you do that heating in an enclosed chamber, the pressure will increase. This would lead you to the idea that warm air is high pressure, but that's not quite a true statement. What is true is that if you heat air in an enclosed space, you will increase its pressure. On the other hand, if you heat the air in an open container with no lid, the pressure wont increase at all: the volume will increase.
Just because air is compressed does not automatically make it "hot." It just means it has a high pressure. If you have experience with compressors, this might not be intuitive because, in practice, you find that compressors heat the air quite a bit (especially if you're filling air tanks for divers!). However, that is really an issue of inefficiency. The piston based compressors take an initial volume of air and compress it really quickly into a small volume. This is going to either raise the pressure, the temperature, or both. In practice, it raises both. That's not a desirable thing -- it wastes money to generate all that heat.
Finally, to your model of collisions, if you have a hot gas and a cold gas at the same pressure, what you will find is that the hot gas will have fewer collisions, but they will be more energetic (higher velocity). The cold gas will have more collisions, but they will have less energy. They're two different approaches to cause the same pressure.