The difference is whether the news is delivered over the air or directly to cable TV operators. The FCC oversees news content on over the air TV because they're using the public airways, a limited resource. Cable TV channels are not regulated to the same extent because they're not using scarce over the air channels.
As far as Fox News being labeled as entertainment, you're confusing the Fox Network which supplies entertainment programming to over the air TV stations which in turn have their own independent news operations, and Fox News Network which provides 24/7 news programming directly to cable TV.
Both were named to reflect their parent company 20th Century Fox, the major Hollywood movie studio.
BTW, all of the traditional OTA networks now have their own cable and/or streaming news channels, with CBS and ABC only recently coming online. These are as free from Federal oversight as CNN and MSNBC (a partnership between Microsoft and NBC).