That's not how hiring works. First you have to get past automated screening, which looks for keywords. Then someone scans a stack of resumes looking for things like experience and a credible overall story. Then maybe you get called for an interview. Reference checks happen after interviewing -- no one has time to confirm details of every resume they get.
The purpose of a resume is to get an interview. That's it. It's not intended to tell all about yourself, or to present a test for the potential employer to pass to prove they are worthy of you. You have 15 seconds or less to make an impression with a resume, so every word has to count, and you can't expect some deep analysis, phone calls, etc. at the first stages.
At most companies the person tasked with turning a big stack of resumes into a short list of candidates to interview is not the person who makes the hiring decision, so you have to get on that short list first.
Anyone can "prove" they had the title Director, that's not the question. What's hard to prove is that the title meant anything or just signals vanity or a two-person company. A Director at Microsoft is a very different job than a Director at Jack at Jane's Design Agency.