Why use 'advisor' or 'supervisor' instead of just 'collaborator'?
I think you're correct that the supervisor role can be equivalent to normal collaboration in that the student will bring their own unique ideas, knowledge, and experience to build on a mutual project collaboratively. However, where I think there is an unavoidable difference (at least, in every student I have met including myself) is the conception of viable projects. In my field I've never heard of a student (masters or PhD) conceiving from the ground-up their entire project. They're always built from a project advertised to prospective students, or developed with the supervisor (often from previous work) where a research relationship already exists.
As a student, you're learning to become an independent researcher. This means learning what is and isn't scientifically interesting and viable, how to search the literature and theory for relevant information, etc.. This is what a supervisor is teaching, and this is a very specific relationship that isn't properly conveyed with &quot;collaborator&quot;.
This distinction is most important when there are multiple co-authors on a paper, including co-authors that are not the supervisor. The non-supervisor co-authors fit the description of collaborators and still should be described as such, no matter their academic &quot;rank&quot;.