Hi Tadej -
Yes, when you check in a page, it is published for viewing. You can also publish a page as is without checking it in, presumably because you are still working on it and don’t want someone else checking it out.
There is no graceful way to check something in without publishing it. You could instead duplicate the page - any new page is essentially “checked out” - and hide the duplicate page in the publishing options and check it in for safety as often as you want. Then when you are ready to publish the page, check out and delete the original, check everything in, rename the duplicate, show it in the publishing options and check it in. Yuck. (I’d test this with non-critical work first - I’ve not tried this method, though it should work in theory.)
I have no idea why the publish to Axshare option exists for team projects. I know that when you do so, replacing the team project is not an option. Maybe it’s there so you can convert it from a team project to a regular one.