Reference request: Reidemeister type moves for immersed curves on surfaces

This isn't true as stated. What is true is that if $\gamma$ and $\gamma'$ two homotopic curves that are in minimal position (i.e. each contains the minimal number of self-intersections in their homotopy class), then $\gamma$ and $\gamma'$ can be connected by a sequence of isotopies and Reidemeister III moves. This was proved in

J. Hass and P. Scott, Shortening curves on surfaces, Topology 33 (1994), no. 1, 25–43.

This paper does not state the precise result I describe above, but see

J. M. Paterson, A combinatorial algorithm for immersed loops in surfaces, Topology Appl. 123 (2002), no. 2, 205–234.

for an alternate proof that states things as I did.

What is needed, then, is a set of moves that take a curve and reduce it to minimal position. Unfortunately, it is not true that you can do this with Reidemeister I and II moves (I am assuming that you meant to include Reidemeister I, by the way), even if you make the obvious modification to allow the curve to pass through the "modification region" in several other arcs that are not involved in the moves. What is needed are moves that eliminate "singular monogons" and "singular bigons" (which can't be assumed to be embedded, as in the usual Reidemeister moves). See the paper

J. Hass and P. Scott, Intersections of curves on surfaces, Israel J. Math. 51 (1985), no. 1-2, 90–120.

for a precise statement and proof (plus examples showing why this is complicated).