Not giving a shit has to do with the level of care given. Not to be confused with "not giving a fuck" see that definition. Not giving a shit means that you literally don't care enough to take action.
For example, Dillon may not want to go to work today. If he doesn't give a fuck, he will still go, but be filled with apathy. If he doesn't give a shit, then he won't bother showing up.
The kitchen staff are not giving a shit anymore so they don't do their fucking cleaning duties.
Simon: Hey, what does not giving a shit about the Third World mean?
Justin: Basically it means you don't care about the problems of people from the most poor nations, duh.
Fogey/fogy /fougi/ sl. (early 18C+, orig. Scot) old-fashioned, stuck-in-the mud.
Person with old fashioned ideas which he is unwilling to change: Come to the disco and stop being such an old fogey!
You think me an old fogeyand an old tory, his thoughtful voice said. I saw three generations since O’Connel’s time. I remember the famine. Do you know that the orange lodges agitated for repeal of the union twenty years before O’Connel did or before the prelates of your communion denounced him as a demagogue? You fenians forget some things. (James Joyce, Ulysses. Penguin Books,1992. p. 38)