Should I host my academic website under my institution domain or under a domain of my own ?
Why don't you just do both? By both, I mean use both URLs. I "do both", so when I graduate, I'll still have my site for others to see. You can do this in many ways, but I had my university student page auto-redirect to my personal home page. The code for that is like a one-liner.

This grants me the opportunity to refer people to different sites depending on the situation. I think myname.com is undoubtedly easier to remember than the nuances in my university student site URL: people.school.org/first.last ... On the other hand, if the situation is more institution-based, perhaps it's better to stick with my college's name. You've got options this way.