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I don't think anything in WordPress actually auto updates without your permission.

Themes and plugins always just give you a notification which links to update :)

I have a break-through.
First, the currently seen problems disappear with an update to current version
Second as the FAQs say, the updates may silently fail if your server is blocking them. apparently, the cURL needs to be set properly.
third, and lots of fun, some WP-servers do NOT manage cURL.

I had to change my site manager package to one that allows cURL. Lots of fun.

Problem solved.

Awesome. Glad to hear!