Positively means in a hopeful way, or definitely and firmly. Think positively, positively identified, positively the last one.
The second sense is an intensifier and has nothing to do with optimism — positively rude means downright rude. In testing, to test positively means the thing was found, which is bad news for a disease and good news for a search.
1adverbhopefulIn a hopeful way.
“Try to think positively about the interview; you're well prepared.”
2adverbpositivelyDefinitely and firmly.
“I positively refuse to go back to that restaurant after the way they treated us.”
positively is a manner, or an intensifier.
Positive plus -ly, from Latin positivus, settled or laid down, from ponere, to place.
Middle English
positively
English, today
positively
From Middle English positively, equivalent to positive + -ly.
Etymology from Wiktionary, CC BY-SA