Li Shenron
Legend
Is it a separate beast? It's just an ability check, and you get to use your proficiency bonus if you are trained in saves of that ability. Which is how all other checks work. The name needs to be there in the same way "arcane check" needs to have that name, to know if you are proficient.
No, it really is a separate beast. If you look at "How To Play" (it's on the second page), you'll see that saving throws are broken out into their own section with no reference to the check rules. If you have something that gives you proficiency on Dexterity checks, it doesn't apply to Dexterity saves.
Neither of you is wrong, really. In a way, checks and saves are similar enough, and become the same thing (a check with ability modifier) for someone non-proficient.
I guess they wanted saving throws to be their own thing because they wanted different characters and monsters to have stronger defense against different effects, beyond the basic ability modifier, but they might have decided that a bonus to all checks with one ability was too much.
Of course, there is also legacy with previous editions to keep in mind. IIRC 4e threw saving throws out of the picture and replaced them with spells attack rolls (or whatever) and that caused serious disappointment from enough people to make them reconsider.