Hello. We have once again come across what people perceive through terminology and what reality is. This is not to say you're necessarily wrong, experimenting with psychadelics, or deprived of sleep. Far from it, in fact.

A sub-domain is also known as a tertiary-level domain name. An example of which is the URL of the website you are reading now: http://forums.hostmysite.com. The association is the relationship of the sub-domain's content relates strongly to the main domain. The forums are a place people can discuss technical issues relating to their hostmysite.com website, or ask questions about websites in general. A sub-domain can be in a folder inside of a website's home directory with a script redirect for traffic, or it can be a separate website entry in the server's web-service configuration.

What most people think of sub-domaining is really sub-accounting (fear not, no bean-counters involved). This is the practice of hosting multiple domains under one site's webhosting space. That's all it is.