There are many online solution which enable me to enter a youtube video url and it download me the video. If I just save the video on my phone and don't upload it on other websites is that legal?
Generally speaking, no, it is not legal. The TOS specifies ways in which content can be legally accessed, and it is only legal to download something that is marked by Youtube as downloadable. I can't say when I last saw such content -- it's not real common. The method that you're suggesting, viz. using some other service that snags material on the fly, is illegal per TOS 4c.