When you start talking calibres things can seem confusing.
308,30-06,30-30,300winmag,300WM are all .308 of an inch.
The .303 is actually .311 of and inch. The 7.62 is .308 still with me

The 30-06 is a 30 calibre invented in 1906.
The 30-30 is a 30 calibre round that WAS propelled by 30 grains (gr) of black powder. 45-70, 45 cal propelled by 70gr
of black powder.

The 6.5x55 and .264 winmag are the same calibre (.264 of an inch)
The 6.5x55,7x57 and 8x57 are 6.5mm,7mm and 8mm bullets with a case length of 55mm,57mm

.410 is a calibre
12,20,16 are gauges

Now the .22 rim fire is .222. The .223,22-250,220 swift shoot bullets that are .224

Mate the list goes on.

My choise for a .22 for rabbits AND foxes would be a .22 magnum. More power for foxes.

Start reading gun mags/books, you'll pick up some good info from them. And sites such as this as well.