if name in ("Jesse", "jesse"):
would be the correct way to do it.
Although, if you want to use or
, the statement would be
if name == 'Jesse' or name == 'jesse':
>>> ("Jesse" or "jesse")
'Jesse'
evaluates to 'Jesse'
, so you're essentially not testing for 'jesse'
when you do if name == ("Jesse" or "jesse")
, since it only tests for equality to 'Jesse'
and does not test for 'jesse'
, as you observed.