Unfortunately the computer I'm borrowing right now doesn't allow me to copy material or even highlight it. It's not state of the art by any means...so you'll just have to dust off those comprehension skills for a moment, if your'e able.
The regulation is set up in such a fashion that it provides additional rest requirements to your benifit if you exceed daily flight time limitations due to unforeseen circumstances. It does not provide relief for exceeding duty periods. In the case of an unscheduled crew for which a duty period is not prescribed, your only requirement at any given point in the day is to look back 24 hours and find 10 continuous hours of rest. Relief is not provided for violating this requirement.
HOWEVER, the FAA has determined that unforseen circumstances may include passenger delays, weather, maintenance issues, etc. Unless you are a scheduled crew with a prescribed regular 14 hour duty day, you have some relief for unforseen circumstances...under some conditions. This is not to say that if you've been waiting for passengers for fifteen hours and they're only moments away, that you can still go.
If you've been on duty for ten hours, your passengers are running late, and you were scheduled to be home in another three, arrival of your passengers an hour from now is unforseen and won't place you in jeopardy of the regulation. Arrival of your passengers four hours from now will, and if you take the flight, not only will you be in violation of 135.267(b) or (c) (depending on your type of operation), BUT also 91.13.
Operators are able to exceed the moving 24 hour limitations (24 hour lookback limitations) on flight time when using a scheduled crew...you have the same duty schedule every day. In any given 24 hour period, you may fly more than your eight or ten hours of flying...just not in a 24 hour calendar day. However, you give something up...the certificate holder can fly you more, but there is no provision or allowance for unforseen circumstances and delays.
If instead you're an unscheduled crew, you're bound to a moving 24 hour lookback period in which your flight time and rest limitations are contained, but because the nature of your schedule is flexible, so is the regulation with regard to being able to exceed your time on duty vs. rest limitations.
Therefore, depending on the specific regulation under which you are operating, yes you can go, or no you cannot...but you haven't provided enough information to say.
Your use of the term "legal to start, legal to finish," is nebulous and therefore vague...ask a vague question, get a vague answer.
You ask if a crew, knowing that their passengers are arriving late and knowing it will get them home over their 14 hour period, are legal to go...depends on the circumstances of the flight and the regulations under which you are operating.
You then ask if the passengers are sufficiently late that they will be departing after the full duty day is over, is legal...yes, and no. I provided you with a legal circumstance in which that very thing occured, and the FAA didn't have an issue with it. I also stated that the circumstances under which one may do this are rare, and unusual. Apparently you found this too onerous for your college brain...try reading it again.
An important aspect of the allowance for unforseen circumstances for an unscheduled crew is that they crew cannot be assigned a trip that would knowingly exceed the time on duty beyond the required rest requirements (ie, 14 hours). In other words, the certificate holder, knowing the weather is bad, the aircraft has difficulty starting up, and the passengers are traditionally four hours late, can't assign the trip (and the pilot can't accept it) on the premise that when these things occur they can be classified as "unforseen" and therefore used as a loophole. Operators do that all the time; it's skating on thin ice, and won't hold water. In each legal interpretation addressing the matter, the FAA always invokes 91.13 to remind those considering these regulations that there is a bigger picture to which they (you) are beholden. Consider it.
Legal interpretations dated 08/30/1993, 07/15/1992, etc, each point out the circumstances regarding when operations involving unforseen delays are legal, and at the same time admonish the reader that 91.13 always applies...even if the flight is legal.
I can't post the interpretations presently, or I would...or perhaps I would for someone else who doesn't cry over the material they're given. Pearls before swine, I guess. You asked, you're the beggar...even a begger can be choosy, I suppose. Try providing a little more information when you ask for help, and then not snapping back at those who offer a helping hand. In other words...grow up.