Here's the SOLUTION I succeeded with right now, using ANDROID 2.3.6
In >Processes (or Applications) select >All
Scroll to "Google Play-Services" >"Deinstall all Updates"
(I suppose this was not the clue, I just like to annex it here as I did so.
I will shurly not go thru all of that steps again to find out if this step is essential.
You may either do asame or try out omitting it)
Now to absolutely all starting with "Google ...", eg. "Google Backup Transport" "Google Calender" etc.:
>"Force to stop" aaaand (here it comes)
>"Data" >"Delete Data"
As done so with the last "Google ..." item immediately a Popup appeared, asking me for the (new) gmail Password.
I thought of the same thing before reading this, after trying every other solution offered from most of the forum respondents, BTW thanks everyone for your advice. Peace out.