On StackOverflow, there's a tag called ”self-hosting“. IMHO it's also (or even more) applicable to serverfault. Here are some quick searches I've done:
See also the Wiktionary definition of ”self-hosting“:
(Internet) Web hosting on one's own servers, without the use of a third party.
What do you think?
My first association with a tag such as "self-hosting" is @HOME, DIY and an amateur approach, rather than managing information technology systems in a business environment which makes it off-topic for ServerFault.
Since questions regarding the polar opposite, shared web hosting, are usually already off-topic and there are already specific tags for most of the large cloud service providers such as amazon-web-services and others the label doesn't add anything for me.
When people manage web servers usually tag the question with the relevant specific apache-2.2, apache-2.4, iis, iis-8.5, nginx and such tags which is usually sufficient that to indicate what stack they are using on their server(s).
IMHO the typical business jargon would also be "on premise" if you really need to distinguish that you're not using a cloud platform.
I would consider this as promoting bad practice (in most situations these days, self-hosting is not a good idea) and a quick look to the questions you linked shows that many of them are quite bad.
