I'd argue that definition 2 is also a decent fit, and definition 1 is acceptable if you permit mild hyperbole. (Which I'd advise doing, as mild hyperbole is more-or-less the default register of Standard American English.)
> mild hyperbole is more-or-less the default register of Standard American English
Well as a speaker of British English, I suppose that's where I fell down.
They're very much not synonyms as far as I'm concerned, my 'exhausting' was a joke, being 'very tiring, causing exhaustion' vs. the 'every possible element, comprehensive' from your Wiktionary link for 'exhaustive'.
You can't possibly check everywhere. I don't say that out of some sort of extinction denial! I assume there are standards in the field for time since sighting over certain number of known habitats or percentage coverage of land or whatever that indicates extinction.
I just wouldn't call that 'after exhaustive search', personally. 'Extensive', sure. 'Sufficient to meet criteria for extinction' is what matters.
It seems believable? I definitely don't intend to accuse you of extinction denialism. But I would believe it's one of those subtle differences between American English and other dialects. I frequently get the impression that, when it comes to range of expression, my native dialect is a bit like music mastering in the CD era: no medium intensity anymore; instead everything's pushed up against - and being clipped by - the upper limit of the dynamic range.
This is a dialect where "awesome" doesn't mean anything remotely like "awe-inspiring", except in books that were written prior to the late 20th century.
https://en.wiktionary.org/wiki/exhaustive#Adjective
I'd argue that definition 2 is also a decent fit, and definition 1 is acceptable if you permit mild hyperbole. (Which I'd advise doing, as mild hyperbole is more-or-less the default register of Standard American English.)