If purpose makes a different then:
I want to be able to archive my travels...mostly for myself. I am not interested in the watch-dog camera on the back and front.
...yea, yea, I 'll probably join all you who have a kazillion miles of video sitting around collecting dust--unedited...may I join your club?
Well in that case then you will more than likely want a camera that is going to give you some good quality video. On top of that you will more than likely benefit with a camera that has image stability..
That cuts down a lot of "budget" cameras, if you don't care about image stabilization or the perfect image take a look at the YI 4K Action and Sports Camera. The YI 4K does have stabilization but its not great. Its a really good budget camera that gives very good quality for the price you pay for it. Its a good entry level camera to start with.
However, since we all ride BMWs we have deep pockets right??? lol Can't really go wrong with the GoPro 7, it has the best video of any action camera currently on the market. It also has some amazing image stabilization with a small amount of cropping. You can go to the GoPro 6 which saves a bit of money, its image quality (now after firmware updates) is pretty solid but not at the GoPro 7 quality.
There is also a new kid on the block that people are raving over, DJI Osmo Action Camera. Once again the video quality isn't 100% at the level of the GoPro, but I'd say its about 95% there. Its very likely that with some firmware updates that this camera will be as good if not better than the GoPro 7. The image stabilization is also AMAZING however it crops the image almost double what the GoPro does. It is a new camera, you can order it directly from DJI or if you want it in a store you have to wait till the 22nd.
There is another post that is discussing mounting points. There are actually quite a few different places you can mount a GoPro on the bike, or you can mount it on your person.
***Just saw someone above mention battery life.. The new DJI goes longer than the GoPros at 4k video, the YI 4k runs about the same as a GoPro. All three of the cameras are able to operate off a battery bank which could give you HOURS of run time. Or you could just use the on bike power and have unlimited run time***