I think he might be wrong these days, but it really doesn't matter. Unless you have a dedicated hosting account, you really shouldn't upload videos to a domain. Shared Hosting and VPS Hosting don't really have the "horsepower" to support more than one or two self-hosted videos. Self-hosted videos usually have problems playing in real time — they'll stutter or stop to preload segments at a time, especially with higher quality/bandwidth videos. Use YouTube or Vimeo to host the videos and embed them in the WordPress site. You don't have the make the videos public on YT, you can simply set them up as Unlisted. In Wordpress you want to embed them with an IFRAME. Something like this:
<iframe src="https://www.youtube.com/embed/youtubeCODE_yt?autoplay=0&loop=0&rel=0&modestbranding=1&origin=https://yourdomain.com" width="720" height="405" frameborder="0" allowfullscreen="allowfullscreen"></iframe>
You can use a straight URL to the video and it will embed just fine, but you don't have any options, most importantly: modestbranding which lets you turn off YouTube logos and (most) references. And "rel" - which used to be set to show "related videos" (that youtube chooses) or turn it off, but nowadays, it always shows related videos, but when set to 0, only from the **same channel** as the video that just played. This is huge! It's basically free advertising!
In wordpress, when you need to enter the iframe embed code, switch the editor to Text to enter it. When you switch back to Visual, you should see the embedded video in your page. (Wordpress may change all the '&' to '&' – doesn't make a difference).
You can definitely upload MOV files to both YT and Vimeo.