I'm sure you'll get many good product recommendations but it really depends on what type of photography you plan to do. There are lots of technical issues to discuss with cameras and you can easily get lost in the specs.
Very basically, for general scenery situations I encounter while trail riding, I've been using an old Panasonic DMC ZS3. It fits in a shirt pocket and the rechargeable battery lasts all day if the flash is used sparingly. It's 12X optical zoom and 10 MP format is adequate for most shots I take. It was a $300 camera when new but I've had it for 10 years now and it isn't worth much should it be damaged or stolen. This is something to keep in mind when on a bike.
Lately, I've been looking at the new $400 Panasonic ZS80 as a replacement. It's 30X zoom and 20 MP format are likely overkill for trail use but I like that it can be recharged via the USB port on my bike.
I also use my smartphone on occasion for certain shots. The quality of the shots it takes is really amazing.
IMO, picture quality, size, ease of use and battery type are the main features to look for in a "bike" camera.
Welcome to the forum and good luck finding the perfect camera!