How best to remove and replace this gasket on swivelling spigot
If using a tool at all, a plastic credit/id/rewards card (use the corner) is the best bet. Tools (such as steel ones) that are harder than the (probably brass) o-ring groove/seat can scratch that surface and cause a leak. 

In most cases you should not need a tool at all. Clean the outer surface of the o-ring with rubbing alcohol and a paper towel to remove any grease on the surface, so you can get a grip on it. Pinch it between your thumb and a finger (on opposite sides of the central metal part), and then move your pinching thumb and finger to one side - this should raise a loop of o-ring out of the groove on that side, which you can grab. You are stretching the other side and pushing up slack material on the side you are moving towards.

The latter method has the advantage of not damaging the o-ring, either - which does not matter when you are replacing it, but does matter when you would prefer to re-use it. 

Clean the whole metal thing above and including the groove very carefully, then lightly lubricate the new o-ring and check that there is no dirt/sand/hair adhering to it, and roll/slide it down into the groove. Clean the inside of the part that goes over this as well, before reinstalling it.