My first association with a tag such as ***"self-hosting"*** is @HOME and other such DIY which is 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 [tag: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 [tag:apache-2.2], [tag:apache-2.4], [tag:IIS], [tag:iis-8.5], [tag:nginx] and such tags which is usually sufficient that to indicate what stack they are using **on the server(s) they manage**. 


IMHO the typical business jargon would also be "on premise" if you really need to distinguish that you're not using a cloud/hosted version