May I ask some experienced users to perhaps create a list of common classical puzzles and common sudoku variants? LMI tests are often so variant-filled that I am somewhat confused what are the originals. Thanks in advance!

You can check Beginners Sudoku Contests to know the common Sudoku Varients.
Other Regular Sudoku tests at LMI contains advanced Sudoku varients.

Adding to what Rajesh said, here is the link for Beginners Sudoku Contest, August 2013 - http://logicmastersindia.com/BeginnersSudoku/?test=B201308 On the right, there are links for the months before that. We've covered a good amount of common variants so far.