Lavalier vs. cardioid mic for low-budget amateur DSLR video recording?
It depends no how far away you are shooting from and how centered your subject will be.  You definitely DO NOT want a lavalier mic for your situation.  They do offer very nice noise reduction while not requiring a boom operator, but they still require setup and distinct micing of each speaker.  This will not work at all in your use case.

You absolutely want a shotgun mic, no question at all about it.  The question becomes, how focused of a shotgun mic do you want?  This is a trade off kind of thing.  The more focused the shotgun mic (moving from cardioid in to supercardioid and then hypercardioid on most mics, but some manufacturers label differently apparently).  The narrower the mic's sweet spot, the less noise you will get, but the closer your subjects will have to be to that sweet spot to be audible.

If you are shooting from a long way away or shooting pointed directly at someone, then a hypercardioid mic is the way to go to get the least interference and focus in the most on that spot.  If, instead, you are going to have a lot of people, all over the frame, and right up next to the camera, then you want just a plain cardioid mic on top that can pick up a wide space of sound, but it will also pick up a lot more noise.

Look at pick up patterns and try to visualize where people will be.  You ideally want them to fall in the coverage circle for the microphone if you were looking top down on the scene.