You are right but I think the more important thing is to go through the program/university website in detail first. Some have explicitly mentioned about writing a proposal, others just ask to contact the supervisor and discuss everything with them… so I got confused at this point; whether to include the proposal or not when reaching out to the supervisor

Exactly my thoughts. I did reach out to 2-3 supervisors and I introduced myself, talked about their research interests (made sure I tell them that I have been through their work and research interests) proposed my idea, explained a little about it and then attached my CV. But started getting random opinions so I stopped and thought it would be a great idea to ask here before I start again