"For example, I might be in PA working remotely on Monday, travel down to DC to work in-person Tuesday through Wednesday, and return to PA to work remotely on Friday and stay in PA for the weekend."
To answer your question about "days", instead think of them as nights.
The definition of an abode is where you spend the night, so DC is trying to figure out how many nights you are in DC:
Monday you work from home and spend the night at home (doesn't count).
You leave Tuesday morning, work in DC and spend the night (1 day), work in DC Wednesday but return to home to sleep (doesn't count).
If you drive to DC on Thursday to work and spend the night (1 day), work Friday and go out partying with friends and spend the night (another day), stay to sightsee on Saturday and realize it's too late to drive home so you spend one more night in DC (another day). Finally go home on Sunday to spend the night (doesn't count) and start again for another week.
You would count one day the first part of the week, but you would count 3 days the second half even if you didn't actually work on some of those days.
DC is trying to find people who stay within the city for more than half the year, and require them to pay income taxes. Many states do this. You won't get double taxed (your state and DC) because your state usually gives you a credit to the tax you pay to DC.
