Two instead of three Reidemeister moves Would there even be any advantage to replacing the type II and type III Reidemeister moves with the following?
Has this appeared in a publication?

One thing lost is type II's ability to link completely unconnected strings, but Reidemeister moves aren't enough to axiomize classic knot theory anyway.
 A: One thing you are always "allowed" to do in math is change the basic rules, and see what you get.  A different example of what you are basically trying to do here is virtual knot theory, introduced by Kauffman.  In virtual knot theory, they don't replace the three Reidemeister moves, but they add a new crossing type and a few moves that take it into account.  This is a sub-field of knot theory, but has its own interesting quirks and theorems that many study without really looking at classical knot theory.  
With that example in mind, what will (probably) happen here is you will find your move is not equivalent to the Type II and III moves.  Thus, you will not have the classical knot theory, but a "new" knot theory.  


*

*What this means for knots:  You will probably be able to find knots which are the unknot in classical knot theory, that are not the unknot in your knot theory.  To do this would be a challenge!  You will need to prove that the knot you have cannot be untangle with your two moves.  In classical knot theory, this is usually done with invariants.  So you will need to construct an invariant which can distinguish the unknot from your knot. 

*On the other hand: This may give you classical knot theory back.  For that to be true, you will need to show that you can get the type II and III moves by only using type I and your move, and vice versa.
If you succeed in either of these, it would be a very nice paper and it sounds fun.  Good luck!
