The definitive information in this case is from the bounty privilege description page, which in answer to the question "How do I award the bounty?" notes:
- You can accept an answer without awarding a bounty to it.
and in discussion of "When does a bounty expire?" clarifies auto-awarding:
If you do not award the bounty within 24 hours of the bounty period ending, half the bounty value will be automatically awarded to the top voted answer posted after the bounty start, provided it has a score of at least 2. If no new answer matches this requirement, no reputation will be awarded at all, and the reputation used on the bounty will be lost forever.
Nothing about acceptance triggering auto-awarding (though there's certainly a case to be made that it should). The documentation the meta FAQ post on the subject appears to be simply wrong, perhaps a misconception from the earlier setup in which acceptance did award bounties. Caveat wiki-lector.
A related and pertinent question is whether the user who posted the bounty thought that they were awarding it to you by accepting your answer, or whether they were purposely using this somewhat arcane feature of the bounty system. They would've received notices about the bounty being near-expiry, but might have just assumed they were superfluous.
The question I linked in a comment above (but didn't read enough of at the time) actually contains a further note from Jeff
suggesting that the wiki-documentation is correct and that you should
have been awarded that bounty:
The implementation has changed a bit to better accommodate careless bounty owners. :)
- the bounty was started by the question owner
- the question owner accepts an answer during the bounty period
- the bounty award period expires without an explicit award
... then we assume the question owner liked your answer when they accepted it, and it gets the full amount of the bounty at time of bounty expiration.
I don't know what to believe anymore.