John's signature appears as "John Bear" beneath his portrait in the 1883 History of Lancaster County by Ellis and Evans. He also used this spelling on title pages of his English-language books, although he printed his German name, Johannes Bär, on title pages of his German-language imprints.
John's sons Reuben and Christian spelled their name "Baer", which is the spelling used for the Baer almanac today. John's biographers have used all three spellings. So do I. I like to keep everyone happy.