So, either choice is not wrong.
C and C++ are clearly different programming languages, so you are actually looking at three languages.
> Rust could also turn into something big, by the looks of it, in the next 10 years.
I wouldn't bet on that. 20 years ago, I heard Haskell and or Java is going to be huge. They are, C and C++ still are. Learn Rust by all means, probably you will be more quickly productive, and it gives you a wide range of choice of targets. That should be sufficient reasons to go that way.
> To me, it seems Rust is so much better than C/C++ in many ways: lots of Python-like abstractions, ...
Yes, but that is also an argument to go for either C or C++. You'll have to learn more. The active code-base in C and C++ is huge, and I don't see everything being rewritten in Go and Rust. Maybe you'll be less productive in churning out code, but it'll give you expertise to look and modify the existing code-base when necessary.
Not that you cannot do that when learning Rust.
In summary, I would go, with Rust, if you want to be quickly able to do your own stuff, C or C++ if you want to be able to work through existing code build "with a hodge podge of welded on features". The deciding factor between C and C++ again should be then, in what language is the code written you'll be looking at. I would say, (and others might disagree) kernel-code (OS kernel, that is), Microcontroller, low level stuff more likely C. C++ you'll find more likely in application code, numeric libraries, etc..