There were umbrella profiled cymbals to be found in the type IIIb period. I had one 22" and there was at least one other I have seen. Same thing for a few 18" I've seen. But in any era it seems that the thinner they get, the more likely (and apparently necessary) it becomes to make it that way. If it was flatter shaped then the wash would overwhelm the stick and render it impractical to play. The deeper profile on thinner one does 2 things. It "tightens" up the cymbal and in doing so attenuates the wash. And the tightness increases the stick definition. But the downside of this design is the deeper profile raises the fundamental up and many of the other tones with it, so that the tonal spectrum as a whole becomes narrower. That makes the cymbal sound "icy". But in cymbals with flatter profiles more range exists between the low's and the highs. But by narrowing that spectrum you can't get much on the low end but the mids and highs become more predominant. Hence that "iciness". Another result is that the cymbal also gets louder in the process, which to my taste is less desirable. With thinner cymbals that combine with a deeper profile, this strategy can actually work, but the tolerance between what constitutes just enough of both thinness profile depth, without too much of one or not enough of the other appears to be extremely narrow, and so consequently the exact combination seems very difficult to obtain. So as a result I've seen it work on only a precious few examples.
I don't mean to hijack this thread to talk about A's but what you just wrote seems to be what I'm hearing with our new 1900g 22" Trans stamp.
As I mentioned is another thread the profile is a bit flatter on one side and a bit more bowed on the other, but the cymbal still lays flat. Very odd. Anyway, I said in that thread that the cymbal sounds very different depending on where you play it. Reviewing it again after reading your post I noticed that on the flatter side the sticking is far less defined and controllable with an almost overwhelming wash if you dig in a bit. On the opposite side which is a bit heavier, with greater bowing, the sticking has much more definition and is slightly brighter, and can be easily heard over the more controlled wash even when digging in. The difference between one side and the other is not subtle. As a result this cymbal sounds its best in a relatively small area.