Terminology
In theory, a camcorder is a video camera that also records video. A long time ago, this was an important distinction, since movie cameras (which are analog and record to film) and other now somewhat obsolete cameras didn't have inbuilt recording capabilites.
(DISCLAIMER: opinionated paragraph ahead!) This distinction doesn't really matter anymore, since most video cameras are camcorders anyway, and/or can record to film and on a digital storage simultaneously (on a professional level). I have really seen the term 'camcorder' used more often to distinguish inexpensive consumer-level cameras from full-fledged, professional video cameras. Since the technology of those aren't fundamentally different (any more), this terminology has lost much of it's meaning.
I learned that a camcorder is a portable video camera that records both visual and audio
I don't know where you heard this, but I don't think that's correct. At least I have never seen it used in that way, but then again, people are mixing up the terminology all the time and especially terms like that that have to some extent lost their original meaning get used in all kinds of contexts and meanings.
Most cameras will record both audio and video. Prosumer-level and some consumer-level cameras will both have an inbuilt microphone as well as an audio jack for external ones. I'm not sure about high-end professional cameras having inbuilt microphones, but at that level you really only use external microphones, since the internal ones have some shortcomings (see External vs. internal microphones).
What do you call a NON-PORTABLE video camera that records visual and audio both?
A video camera. Or call it whatever you like, it doesn't really matter.
External vs. internal microphones
As mentioned, most cameras on a prosumer-level (which seems to be the right type of camera for your purpose) will have an inbuilt microphone and an audio jack to record with an external microphone. I would definitely recommend you use a good microphone. So many Youtube channels are unwatchable because of their terrible audio quality. And while even low-cost cameras have pretty good video quality nowadays, the audio quality is usually very poor.
Internal mics suffer from some problems. For instance, they are by design built in to the camera, so in most cases they won't be close to the source of the sound. Combine that with the fact that they are usually omnidirectional (i.e. they record sound from every direction) and they are generally not of high quality and you will get noisy, unclear audio with much reverb and ambient noise. Poor sound quality is one of the first things that people will notice about your video, and at least for me it's a reason to quit your video and never come back.
Your questions sounds like you plan on recording your videos at home. In that case, I would recommend a solid desktop microphone (that is one that has it's own stand). Depending on what microphone and camera you get, you might be able to plug it into the camera and use it as an external mic for your video recording, but you will probably have to connect it to a computer and record your audio from there. While editing, you can use the wave form of the camera's audio or automatic matching to match your external audio recording to your video.
