I'm not sure I follow. For the purposes of my example the firefighter has no legal recourse if you don't listen. They're just random volunteers where I live.

I don't want to get too hung up on definitions because that's counter productive I think. So what I'm talking about is that sometimes humans rely on power, real or perceived, in order to demand that others subsume their own desires and submit to those of the powerful.

Examples are police and other violent gangs - do what I say or I shoot you, capitalists - work for me or I will starve you, shitty parents - do what I say or I will hurt you.

I am calling that authority, notice that at no point is there consent from the person authority is being claimed over (it's not consent if it's coerced).

On the other hand people sometimes agree to perform certain roles with each other, or to be bound by certain rules in order to undertake some endeavour. For example when I am teaching my niece science she agrees to solve the problems I ask her to solve, but there is no coercion here. She is free to say at any moment "no" and I am free to either withdraw my offer to teach, ask a different question, propose a break or whatever else. Similarly working groups might elect someone among them to manage a project, but this isn't authority (as I have defined above) if they are free to relect a project manager, refuse directions or whatever.

Various writers have waffled to varying extents trying to pin down specific definitions. I side with those who think it's clearer to distinguish between the two social arrangements by not calling the second one authority.