I am glad to learn that you are a German who uses the "leaky steam pipe" "ich", as you so eloquently put it! Usually, German-speakers here (because they learned that way) will say "ish bin", which I think sounds a little coarser than your own, softer, pronunciation at the back of the palette. Which indeed was how I was taught at school (I presume it's a regional dialect thing).
Thanks for the other information, too: I didn't know any of that.