This is a 10 speed chain, driving a 9 speed mountain derailleur, on a 10 speed cassette, with an 8 speed crank and a 9 speed road bike front derailleur (for doubles, not triples). With road shifters.
Huh, thought I had another picture of the current bike. Nevermind, use your imagination. Point is, the differences between the physical dimensions of the different spacings is so small that you often need to go two or three generations apart before you even have the beginnings of problems. You wouldn't have issues with the chains until you, say, tried to use a 9+ speed chain on a 5-6 speed freewheel.
Many problems aren't really problems, so much as they are things to be aware of when you are assembling the bike. For instance, chains are 3/32" [nominal] internal right up until 11 speed, so you're fine on that front--only the outside slims down, but you're talking about fractions of a millimeter. Technically, you'd be hard pressed to use a 10 speed chain on a single speed 3/32 sprocket, but for derailleur purposes, and their narrower teeth, they'll fit even 7 speed HG-style sprockets.
Even so, you'll only run into issues trying to use a modern flush riveted chain (9 speed+) with early indexing systems. They'll work fine, if sluggishly, on friction systems.