"by exam" or "by exams"

I'd use 'by exams' because students take more than one exam.

Subjecting students to tests or examinations is only one of the many educational methods.

But according to the rules when used to say what means or method someone uses to do something,

by-phrase shoud be by+singular form noun without "a" or "an". eg.
You can reserve the tickets by phone.
Send it by airmail.
Some customers prefer to pay by cheque.
by car/train/bus/taxi etc
They travelled to Chicago by train.

And then, how to explain it? Thanks.

Some customers prefer to pay by cheque. - Cheque is the common means of payment for these customers and since it is common for all, you need not use the plural
noun.
by car/train/bus/taxi etc - The different means of transport are individually a common one for such category of travellers