Actually, I think that in the case of your junction all the data's already in OSM. You just need to create a map that uses it.
I'm guessing that so far you've downloaded a map for your Nuvi, perhaps from somewhere like [http://garmin.openstreetmap.nl/][1] (although there are [lots of others][2]). Most of the maps at these download sites will have been created using [mkgmap][3]. That's highly customisable - you can choose which things from OSM are used in the map on your Nuvi and what they are named, but in order to do this you'll need to create the map yourself. The way that customised maps are created is by using a particular mkgmap "style" - a folder of files describing how particular items should be processed.
You mention that your Nuvi knows the road as "Keizer Karelweg (S108)" - a combination of the road "name" and "ref" of [this road][4]. The lines in the default "lines" style file that control how name and ref are handled are these:
# Set highway names to include the reference if there is one
highway=motorway {name '${ref|highway-symbol:hbox} ${name}' | '${ref|highway-symbol:hbox}' | '${name}' }
highway=trunk {name '${ref|highway-symbol:hbox} ${name}' | '${ref|highway-symbol:hbox}' | '${name}'; add display_name = '${name} (${ref})' }
It looks a bit complicated, but you can use the documentation on [this page][2] to help understand what's going on.
What I suspect that you'll want to do is to create a rule for "motorway_link" that incorporates "destination" as well as "ref" (as in your case "destination=Amstelveen" is [set on the motorway_link][3]). I suggest that you experiment with the style rules using a small area first (to save time when creating test maps and copying them to your Nuvi)
[1]: http://www.mkgmap.org.uk/doc/index.html
[2]: http://www.mkgmap.org.uk/doc/index.html
[3]: http://www.openstreetmap.org/browse/way/7653701