Levenstein distance and dynamic time warp
There is in fact a whole bunch of related algorithms. In modern context I believe "time warp" would be called sequence alignment. Depending on whether you want to match complete strings or optimal substrings one gets Needleman-Wunsch and Smith-Waterman. 

In your latter algorithm the costs seem to vary, that is one can attribute different costs for deletion and insertion of a character, as well as for changing characters. Your first algorithm seems to fix these costs to $1$ for all three possible changes.