Ok, so I think I mis-interpreted the OP...it states:
To arrange my DVDs with precision I will take one DVD off the shelf and place it in the correct spot back on the shelf.
If that's the case, let's look at the example of 2, 4, 1, 3:
Start with 1...move it to its correct spot (and shift as needed): 1, 2, 4, 3...then try 2...it's already there...then 3, and move it...1, 2, 3, 4...you're done.
However, you can also do this by moving the 2 between 1 and 3 initially:
2, 4, 1, 3 --> 4, 1, 2, 3 --> 1, 2, 3, 4...
Which of the above is what you would expect from your OP? Ei