If you don't wanna read all this, just click the second youtube link, that'll send you to the moves of the best solution.

There are conceptually two solutions to this puzzle, where the second solution is the first one played in reverse.

I will mostly reference the top and bottom image of the following link: https://imgur.com/a/khnXlY9 as Ref 1 and Ref 2

The easiest way to think about this Puzzle is by drawing which tile is connected to which. If you look at the top of Ref 2, you see the puzzle in Checkerboard form on the left, and as a graph of connected fields on the right.

Solution 1: The solution described in Die4Ever's all puzzles solution is conceptually what you can see here, just a little optimized:

At the end of the puzzle you have 2 choices for the final 4 moves, which is worked out in the bottom of Image Ref1. It turns out the version in the run shown above is optimal.

Solution 2: The other solution consists of going backwards through solution 1. Since solution 1 has 2 options at the end, this means solution 2 has 2 options at the beginning, worked out in the top of Ref 1.

The option shown here is the faster of the two

Finally, Ref 2 compares the best paths, forwards and backwards. Both have the exact same number of clicks, animations ( = moves), and choices (which-way-should-I-go-now's). However the solution shown in the second video has 1 fewer mouse movement. This timesave of less than a second was figured out in a 90 minute session of a stream. Totally worth it.

Ref 1 & 2 are the working out of the proof that that is the fastest solution.