The video will stream from wherever it is served. Other than the watermarks created by the various services like YouTube the video will appear to live inside your page if it is added to your page with embed code. The only other way is to link to the page on the YouTube site but that sends users away from your site. So, I think the watermarks makes it clear. If there is no watermark you could put text below the video indicating where it comes from if you wanted to.
There is nothing you can do to make YouTube re-route users to your site. You can create videos that have your own watermark in addition to the YouTube watermark. I think you would have to do that before uploading it to YouTube. This way if users find your video on YouTube they will see your site url in the watermark and that may drive some traffic to your site. So this is a good way for people to find out about your site since YouTube gets a lot of traffic. You would want to tag your videos appropriately on YouTube so people searching on tags find it, not very video uploaded to YouTube gets a lot of views. YouTube provides some premium services to help you promote videos though. Another video site worth looking at is Vimeo.
Hope it helps,