Do you believe you have two thumbs?pic.twitter.com/AOSFcGS1Au
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I need to do some empirical work on this; my anecdotal understanding may be mistaken.
"believe"' implies greater emotional investment, such as ideological commitment (seems heavily influenced by the Christian tradition). It suggests "ought" more than "is". But you can cue this much more strongly by other word choices like "will" vs "should".
I would not say "believe" implies greater certainty, if anything the anxiety in it suggests the plausibility of doubt. Perhaps my atheism is showing.
I don't think your understanding is mistaken. There is a reason you can use "believe in" (and that it is infinitely stronger than "think that", and there is no "think in") and that "a belief", connotes great weight and meaning, but "a thought" suggests triviality.
Also, @SomeDrake's example is about a future event - yours is about a present event ('having two thumbs'). To the extent that the terms are 'coextensive', I'm pretty sure this is mostly only for things in the future.
Just thinking about it thought I'd check across 1st, 2nd, and 3rd person. Results are the same (using "that" which is more common): https://books.google.com/ngrams/graph?content=I+believe+that+I+am%2CI+believe+that+you+are%2CI+believe+that+they+are%2C+I+think+that+I+am%2C+I+think+that+you+are%2C+I+think+that+they+are&year_start=1800&year_end=2000&corpus=17&smoothing=3&share=&direct_url=t1%3B%2CI%20believe%20that%20I%20am%3B%2Cc0%3B.t1%3B%2CI%20believe%20that%20you%20are%3B%2Cc0%3B.t1%3B%2CI%20believe%20that%20they%20are%3B%2Cc0%3B.t1%3B%2CI%20think%20that%20I%20am%3B%2Cc0%3B.t1%3B%2CI%20think%20that%20you%20are%3B%2Cc0%3B.t1%3B%2CI%20think%20that%20they%20are%3B%2Cc0 …
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