What's a good mental model for what it means that the flow of time isn't constant in general relativity?

General relativity says that, as I move my finger around in my office, through Earth's gravity gradient, the "flow" of time at my finger tip is constantly changing (though by very small amounts, ~10²⁰ or so.)

What I haven't been able to understand is what does that change in flow mean when you try to understand time. In other words, the usual mental model is that time flows the same for everything in your office, say. But that's not correct! Time flows slower for things closer to the floor than to the ceiling (unless you have a really large mass hanging over your head...)

Another way to ask the question is: ok, suppose I have a small charged black hole floating in front of me. It's charged to hold it in place. Let's assume it's big enough that the Hawking radiation isn't enough to worry about. I stick my finger near it (but not too near.) The tide is bearable. I can pull my finger out if I want to. Now time is flowing slower at my fingertip. What happens? Can I feel the difference in time flow? Does my finger appear red-shifted?

I know it's a kind of fuzzy question, my apologies. While I can easily work out the math, it seems clear to me I still don't really understand what variable time flow MEANS.