It's my opinion that how you manage career transitions, forgive me for using an unnecessarily gender-specific idiom, separates the men from the boys when it comes to professionalism.
The ideal is that you use things like exit interviews as a canvas on which to have a broader discussion about both career and business priorities that will leave them very interested in bringing you back on in an expanded capacity once you've grown in your career more.
This requires the ability to clearly understand and distill why you couldn't get what you needed out of your current job. Nobody's really expecting you to be unhappy, but at the same time, there's a kind of expectation that you raise issues and work them out before they reach the point where you go find another job.
Naturally, something like compensation isn't part of this expectation, if another company is offering you $30k more, you won't be expected to have asked for that before handing in your notice, because obviously you weren't going to get it. But if compensation is the primary reason you're leaving, then rationally speaking, you should be open to accepting a counter-offer. If you're not open to that, then you need to clarify more reasons why you're leaving.
My understanding is that it's best to tie what you want and why you couldn't get it to the business model of the company. That offers the least-offensive pathway through an exit interview that results in no legal issues and the utmost in professional candor. They can't change the business model, so another job is a perfectly appropriate pathway to getting what you want. This naturally requires you to understand the company's biz model and this is best accomplished by having discussions with your manager.
If you want career advancement, doing it through your current job requires you to wait for an opportunity to open up. I would have happily stayed at my last job if I could have led a team, but that wasn't in the cards. But it was an option that we explored.
If you have a personal problem with someone working there, then the expected resolution is to restructure the teams so you don't have any interaction with them. This one is a bit tricky, I wouldn't bring it up in an exit interview, but behind closed doors long before you start looking. This way you can simply hint at it in the interview and everyone knows what you mean without having to actually discuss it, while the surface discussion is about business priorities. A personal problem affects team cohesiveness though, and if management isn't willing to resolve them, then that points to a political problem with the organization.
Anyway, I hope this offers some frameworks and tools to help with career management.