Interview questions you don't like/fail shouldn't be viewed as reflective of you, but reflective of your fit at that company.
They asked a puzzle-y question you couldn't solve? Either their problems are puzzle-y, or they have terrible criteria for evaluating people; either way, do you really want to work there?
They asked a hard algorithmic problem, the kind that was someone's PhD thesis 30 years ago, and which you'd only get the optimal result to if you'd seen it before? Either they really do expect you to know stuff like that off the top of your head, or they expect you to behave a certain way when confronted with stuff you don't know; either way, do you really want to work in an environment with those kinds of expectations?
They asked you to code something hard withing a short time limit, and while you can code it, you failed the time limit; do you really want to work somewhere where speed is -the- criteria to optimize, rather than quality/maintainability/whatever (or at least where they're selecting for that rather than those other attributes)?
Etc.
In general, as others have said, if you program, you're a programmer. What specific roles you're a fit for is largely orthogonal to that.
If you don't pass a company's test, view it as having dodged a bullet, not having 'failed'; either they expect you to have something from day one that you don't have, or they are terrible at testing for what they really want, and so their culture, codebase, etc, is likely miserable. In neither case do you want to work for them.
You want to work for a place whose expectations for what you have on day one fits what you actually have (whatever they may be), -and- whose interview process tests for that (even if it's a little flaky), so both you and they can expect, going into the job, that you can handle it.
Remember, too, that there are false positives and false negatives even when they're testing specifically for what they want. Failure to figure out a puzzle doesn't even mean the person isn't good at figuring out puzzles; just that in this case, in this contrived, high stress set of circumstances (an interview), they were unable to come up with an answer in the time limit.