I can't imagine why anyone, military or civilian, would object to being described as a first responder. They all rush in to help others, often when danger still exists. As far as I'm concerned, "first responder" is one of the very most honorable titles any person can have and it certainly takes nothing away from any other titles or job descriptions that apply!
It's an interesting term. Very American. Not used anywhere else until recently.
It's always difficult to find the history of a term, but I can't see it used in the Australian media before 2011, and there's no use of it again till 2014... And it took off in 2015.
To me - and *please* this is just an opinion about language - there's several words or phrases that become used or misused because of their emotive effect. "Hero" is a classic. Now it means little or nothing. Yesterday there was a report of a Hero baby that underwent a heart operation at 1 week old. Extraordinary surgery but the baby had no choice, no heroic deed done by it. There may have been heroism or just vast risk by the operating doctor... But not the baby.
"First Responder" in Australian vernacular, not American, is a bit weird. Who is that? Anyone with a badge? Like Stephen Crane's The Red Badge of Courage it can mean sweet F a. The person is paid to do a job and happens to be there because of a rota, luck, on duty etc etc. There's no special 'thing' to being employed.
In Australia the largest Australian flags are flown at used car dealerships... In the USA the largest mobile US flags are flown behind firetrucks.
Cultural differences can not be easily understood by travellers. But it's very important to be able to identify these differences... That's education ☺️
That we use language differently dies not make it correct or incorrect... It's just different. (recent example America being a Republic / democracy / both etc. Those words have a different meaning outside the USA. Surprised the hell out of me)