Well, the website you quote certainly takes you to the right place, which operates as I described. I just tried it: it doesn't ask you for your reg number, and it doesn't ask you for the make!
The site linked from there which
does ask you for those things is "Vehicle Enquiry", which is used when you want to check details of a vehicle, such as its date of registration, its emissions, engine size, and when the tax is due. (Incidentally, I tried this "Vehicle Enquiry" for a Messerschmitt of which I happen to know the number. The list of makes didn't include Messerschmitt, but I chose "other..." from the list, and typed "messerschmitt" in the "other make" box - and up came the details, no problem.
It's "Apply for a Tax Disc" that goes to the right place.