You have to fight back against this kind of thing. You have a big advantage, you can easily leave and they can't easily replace you. So use that to extract concessions in the form of salary, work-from-home days, latitude in which tech stacks to use.

I suspect the jobs that tick ~2's boxes are more likely to be jobs where managers see programmers as cogs in a machine. You don't have leverage if they don't believe they can't replace you, even if they can't. And in that sort of job, I think it's quite a bit more likely that they can.

You have to demonstrate the reality of the situation to them before they'll come to see it your way. In this case, the most effective thing would be to go get yourself another job offer, print it out, and bring it into your department head's office. Say you kind of like it here but you can't ignore the $X000 extra you'd be making over there.

Then negotiate. The job offer is your leverage. It says, directly and to the point, that you're an undervalued asset, and the fact that you're in their office states that you're willing to deal with them to make sure all parties are satisfied.

Until you bring this document to them, then there's no reality anywhere in the equation and all that matters is what's being perceived. Since they're paying you, they will rightly believe that their take on the situation should be considered the correct one. Right not because it better reflects reality, but right because the balance of power at the moment favors them.