"In general,...": do mathematicians use this phrase oppositely from everyone else?
define:general gives,
affecting or concerning all or most people, places, or things; widespread
I suppose Mathematicians are interested in whether a rule is always true (for "all" things), whereas other people are satisfied to be interested in rules which are usually true (for "most" things).
Note that these are not (as your question alleges them to be) "precisely the opposite connotation".