Cyclist (and the almost never used cycler) refer to bicycle riders. I suppose you could apply them to uni- and tri- cycle riders, too.
I see the dictionary links show it can be applied to a motorcyclist. I rarely hear it used that way, and until I googled for an example to show you, I didn't realize it had any prevalence. From what I'm seeing, it seems to be used to mean motorcycle rider only in headlines . . .
Example of motorcycle rider and cyclist in typical (to my ear) AmE.
Contradictory evidence to my own statement, here. You'll note that the headline uses the word cyclist, but the text says motorcycle rider.
There is an American Motorcyclist Association, too. So I would throw that terminology on the pile, too.
Bike rider is generally a bicyclist. While bike can be generalized to mean any two wheeled vehicle (including motorcycles), no one would confuse the meaning as anything but bicyclist.
Biker is the only ambiguous one. While it conjures up images of sweaty, tattooed men with beards dressed in denim and leather, it can also be applied to bicyclists (although cyclist is more common than biker for this).
Typically when referring to motorcycle riders as bikers you are referring to people who live the lifestyle. To risk stereotyping: they dress the part, get tattoos, and are often members of an MC (motorcycle club). You don't usually refer to a casual motorcycle rider or someone who commutes by motorcycle as a biker.
That said, not every "biker" is a member of a gang, deals methamphetamine, kidnaps women to sell into white slavery, etc. Many are just people who enjoy the outdoor, open road lifestyle. And, many are "weekend bikers," who have run-of-the-mill weekday jobs, but, spend their weekends cruising on the roads with an MC of like-minded people.