What is so preposterous about consulting a dictionary, actually several dictionaries, about the usage of a word? And in fact I did ask if my perception was right or wrong.
About "denumerable" Merriam Webster says "capable of being put into one-to-one correspondence with the positive integers".
The American Heritage Dictionary says "Capable of being put into one-to-one correspondence with the positive integers; countable".
The Encarta Dictionary says 'countable: able to form a one-to-one correspondence with the positive integers [Early 20th century. Formed from late Latin denumerare “to count out,” from Latin numerare “to number” '.
Synonyms for "denumerable" have included enumerable, countable, and numerable.
None of those definitions or synonyms hint that "denumerable" and "infinite" are kin if not quite the same. The specialized terminology is used by mathematicians and transfinite theorists, a specialty that I am sure few of us, including me, have ever heard of before.