There a many ways to break a fishing chain.
When we talk about moving, we are talking about the changing between the control pads (circle pad and the standard 4-direction pad/D-pad).
If you change direction using the circle pad, you probably know that it activates the roller blades which changes the overall atmosphere - making you lose the chain
If you change direction using the D-pad without moving, it is still considered as changing fishing spots as you look away from your previous spot (still a form of moving). So I believe the answer to your question is YES
Hope I helped! :)