This sounds right.
A major theme: remove the defender.
I'm now doing one hour of tactics per day and hoping to get the patterns into my brain. I was doing it quickly, with only 50% accuracy, but after reading here, I am slowing down, forgetting the timer, and looking for accuracy.
Every so often, a clunker arises. I try it against the computer and the computer does a different move than the tactic!
I first look at what pieces are protected and unprotected.
Next, I look for familiar tactical patterns (pins, double-attacks, checkmates, etc.). [Of course, if a pattern is super simple or drilled into my head very deeply, then I may notice this while looking for protected/unprotected pieces.]
Then I check my calculations before deciding on a move.
How do other people approach things?