Since big-O and other asymptotic notations come from calculus I don't think it is a bad idea to know precise definition. For programmers it is questionable how useful they are in everyday job and I would just forgo asking them during interviews, although I have been asked in recent interviews. These asymptotic notations are immensely helpful in finite approximation of continuous functions using infinite series like Taylor series or orthogonal polynomials. Computer scientists borrowed the notations and abused them as if they were the final arbitrator of things when they are not; Knuth noted too much game is being played with them in academic publication. Further attempts at drawing analogies different kinds of convergence and growth functions have not found widespread use.

I guess I am saying don't underestimate the importance of rigor in knowing precise definition of mathematical notations.