That’s all very fine and good, but how do you actually define the perspective of that observer? A short, more pedantic answer is that the observer determines the coordinate system. If you want your “left direction” to become your “forward direction” you just rotate 90° to the left. So to define an observer in relativity, you define a coordinate system with them at the center and a time defined by what their clock says. When someone is moving, they’re moving relative to you and according to your coordinate system.