Wow, regex101.com is very nice, but I never would have noticed that debugger pane if I hadn't gone looking for it after your comment. Incredibly useful tool.
No, that would work very similarly to the greedy version. The backtracking happens because the \d gets matched to the '1' and the whole thing has to be rolled back when the $ attempts match and instead finds '2' (this would happen again if there were more digits for \d to speculatively match on). So the backtracking is not caused by the laziness or greediness of the \D* ; we really do want to gobble up all of the non-digits.
On the two options generally:
/(\d)\D*$/
is problematic if you have a lot of digits, while
/.*(\d)/
is problematic if you have a lot of text after the last digit. Both could potentially be optimized by the engine to run right-to-left (the former because it's anchored to the end and the latter because it greedily matches to the beginning), and then both would do well. I'm not sure if that happens in practice.
Overall, I prefer the latter, both because I think it's clearer and because its perf characteristics hold up under a wider variety of inputs.
Edit: how do you make literal asterisks on HN without having a space after them?
Technically, while they both capture the same digit, the match itself it different, including either everything before that digit or everything after it. But I tend to liberally use lookaround to keep the actual match clean myself; maybe others go more often for a capturing group. (Well, and not being able to use arbitrary-length lookaround in most engines might be a reason too.)