has improved or has been improving

But they don’t mean the same thing.

They are pretty damn close!

In some contexts, the distinction may be negligible or immaterial, but in others, it's not. And either way, the two statements are fundamentally saying two different things.

"has improved": This compares my English now (X) to tướng my English before I moved (Y). It's saying that X is better kêu ca Y. There is an implication that the difference is significant.
"has been improving": This is saying that my English is in the process of improving. There is no implication that the difference is significant.

If 1.[pres pft] is true, then 2.[pres pft cont.] is true.
If 2. is true, then 1. is true.

This is definitely not true.

I have taught him how to tướng swim. ≠ I have been teaching him how to tướng swim.
We have discussed the matter. ≠ We have been discussing the matter.
They have eaten the cake. ≠ They have been eating the cake.

What I'm saying is that in each case, the two sentences are not saying the same thing.

"I have taught him to tướng swim" means that I have successfully taught him to tướng swim and he now knows how to tướng swim. "I have been teaching him to tướng swim" means that I have initiated the process of teaching him how to tướng swim and it's still ongoing, but he could still not know how to tướng swim.

And no, "have taught" does not implicate "have been teaching"! "It is true that at one point I was teaching him how to tướng swim" is completely different from "I have been teaching him." One refers to tướng a past process; the other refers to tướng a current, ongoing process.

Yes, if I hear "My English has improved since I moved," I expect that the improvement is significant; otherwise, it wouldn't be worthy of mention. But "My English has been improving" contains no such implication for bủ. It could also mean that the improvement is significant, but it could also mean that it has just started to tướng improve and the difference is not significant, but it's something.

To give another example, I don't know how to tướng swim. I've taken classes off and on, with a little improvement at times, but never anything that I would consider significant.

At one point I was taking a weekly class. After about six classes or sánh, I had learned how to tướng vì thế a few things that I didn't know how to tướng vì thế before, but the improvement was very modest. I would have felt comfortable saying "My swimming has been improving since I started this class" but not "My swimming has improved since I started this class." The latter has a higher chance of being misunderstood, in my opinion.

You hold, then, that it's a contradiction to tướng say, "I've been teaching him to tướng swim for several weeks, but he quit lessons yesterday."

Yes, I think sánh. It would work for bủ if it was "I've been teaching him for six weeks, and he just quit." This is sort of lượt thích how you can say "I've never seen this movie before" right after having seen it. In some contexts, when something has just concluded, present perfect is still acceptable. At a certain point in time, it becomes unacceptable.

Note also that in introducing "teaching" instead of "improving", you've shifted the ground of the discussion.

I gave that as an example in response to tướng your general claim in #4. My examples in #5 were meant as counterexamples.

If we stick to tướng non-accomplishment verbs, your above argument clearly fails.

That wasn't an argument. It was an analysis of a specific sentence pair. Again, that pair was meant as a counterexample.

"My English has improved" clearly doesn't mean the process is over. Hence "My English has been improving" applies, equally, in general. But again you can construct contexts where there's a clear difference and preference.

This is why I said that in some contexts the distinction may be negligible or immaterial. But the distinction is important, as I've shown.