I also use a P-Touch label maker. It makes excellent, readable labels which do not come off but which I can easily remove if I need to. The P-Touch labels can be purchased in normal and wide widths as you prefer. And the labeler can be adjusted to different fonts, all caps or not, and borders can be added around the name or title if you wish.
I find that also covering a printed label with a clear label (not clear tape, but pretty much the same thing) either round or rectangular, holds the other label in place for years. A round clear label may work best since it has no corners, and it's the corners that always seem to lift up and peel off first. I get them on
Amazon in rolls of round clear labels which should last years at the slow rate I'm using them.
As for the way you label your binders, it might be easier if you listed each country in that binder on a separate label up and down the spine. Then, when you have to move a country to another binder, just remove its label and make a new one for the new binder.
I also find it's a lot easier and quicker to locate the binder for a country by skimming the list of country names than it is to "do the alphabet thing" to identify if the country I want to locate is between the first and last country listed. But maybe I struggled with the alphabet as a child . . . . Is "Austria" before or after "Australia"? I know this isn't difficult, but my brain hesitates -- while just seeing "Austria" in the list on the spine makes it quick and easy to find. Most binders will only have a few countries in them, so there wouldn't be many long lists on any spine.
Have fun!