I would agree, to the extent that terminology may matter - it ideally wouldn't.
This scenario OP sketches sounds an awful lot like it can be extrapolated to 'no matter whether I try and fail, or don't try and fail, I'll fail anyway. as such, why even try?' which may indeed match the hopelessness associated with depression more, which may itself trigger more acute anxiety when trying anyway.
I think it matters because the treatments and prognosis for anxiety and depression are usually quite different. For example, the mindset if you have long-term anxiety disorder should be "how can I treat this?" but the mindset if you have long-term depression (dysthymia/persistent depression) should be "how can I cope/live with this?"
I definitely agree with your second paragraph. I would actually extrapolate it slightly further and say that it's worth OP asking themselves if they actually simply feel unmotivated, and anxiety is a good way to pin that fact on something tangible or understandable. If my hypothesis that it is a depressive disorder is correct, I think that is quite likely. But that's a question for OP, obviously not one that I can infer.
This scenario OP sketches sounds an awful lot like it can be extrapolated to 'no matter whether I try and fail, or don't try and fail, I'll fail anyway. as such, why even try?' which may indeed match the hopelessness associated with depression more, which may itself trigger more acute anxiety when trying anyway.