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As far as I can find, the FAA doesn't define the term landing so it could be argued either way. The "typical" definition of landing is:

1: an act or process of one that lands; especially : a going or bringing to a surface (as land or shore) after a voyage or flight

That definition doesn't say that it has to be "controlled" or that the airplane has to be usable again! Even if you decide that it wasn't a "successful landing", you could always get your third landing later if you are still allowed to fly, and then carry the passengers.

So this brings us to the second part of your question: Are you allowed to fly after an accident?

Since you had an accident (as defined by 49 CFR 830) you must immediately notify the NTSB of the accident. Once you do that, you have 10 days to file the required report.

At this point, you still have a valid pilot license and can legally fly another airplane.

Chances are however, that you will be receiving a certified letter from the local FSDO requiring a reexamination in order to keep your license because your competence has been called into question. If you don't call them to make an appointment, they will start the process to suspend your certificate. During this time, you can also continue to fly because you still have a valid license.

The FAA can also use their emergency authority to immediately suspend your license if they feel that it is warranted. This can be appealed to the NTSB board if you feel that it isn't justified, and they have up to five days to respond. At this point you are not allowed to fly pending the response from the board.

This, and a lot more information can be found on AOPA's FAA Enforcement Webpage.