All of these are excellent but, obviously I would much rather study at Hong Kong Polytechnic University than any other others... world class city, world class education, and a world class experience...
Other than the courses, you have to teach yourself and become disciplined at asking the right questions and giving the HONEST answers. Design is not just about how something looks, its performance, ergonomics, functionality, manufacturability, logistics (size, weight, deliverability, assembly knowledge, age group, etc.) and cost. You really have to understand EVERYTHING about what you are doing. There are lots of designers that are given a task and they design a product but, its really a team effort and that is required when you are working at a company and following someone else's vision, but when you are the head of the company then everyone is following YOUR VISION. You need to be adept at everything - that doesn't mean that you do everything but, you should know enough that no one can bullshit you because if you don't you could be headed for a train wreck... best of luck, develop your skills, and use common sense.
I do not know the opportunities in India or the nature of what is expected from mechanical engineers their. I am now retired and over the years I found that mechanical engineers with the use of 3D cad were doing the jobs taken up by drafting persons or moved into finite element analyses. Other mechanical engineers when into supporting manufacturing or civil engineering. The opportunity to do design is related to the nature of the company and where your job is. The the people that can best meet that need in the end will get the majority of the design work.
I worked for an aerospace manufacture of a generator and other systems. They hired many electrical engineers from India because of speaking English. I supported them as a packaging mechanical engineer. The company hired Indian professionals in the math based fields.
I also worked for medical and consumer product companies with fewer but still Indian engineers. English opens many doors.
I have about three dozen patents with a masters degree but the degree wasn't the reason for the patents. It is the ability to solve problems that are outside the experience of the other engineers or the company. Education doesn't teach how to think and discover solutions. I never created from scratch but altered solutions from other industries.
So the first question you need to answer is where do you excel. I can assure you that if you excel in design they will make use of your capacity. Get a job in a company that need lots of design work.
no, they are not separate. If you go to a university in the usa you will get a master of science degree and it will show your area of study as being whatever engineering you are doing. "ME" is usually short for "Mechanical Engineering"