I think this conveys the right intuition but it's a bit finicky and artificial to make that concept precise so I think it's better to say that the concept of mo

I think this conveys the right intuition but it's a bit finicky and artificial to make that concept precise so I think it's better to say that the concept of moving through spacetime at a given speed is not well-defined. Particles have a trajectory through spacetime. They don't move along the trajectory because the time dimension is already part of the trajectory. So the speed at which they move along their spacetime trajectory seems to be a meaningless concept. What is true however is that if the particle takes a clock with it, and you mark a tick every second the clock ticks, then the ticks will be equally spaced along the trajectory (where distance is measured using the Minkowski metric). This makes a bit more precise what it means for photons to be stationary along the time dimension. It does not mean that they move horizontally in a space-time diagram. They move diagonally at speed c, as you expect, so for an outside observer they do move through time. However, the Minkowski distance between any two points on photon trajectories is zero. So if you wanted to annotate their trajectory with the clock tickmarks you could start somewhere and put your first tick there, but then your second tickmark would be infinitely far away on their trajectory. Thus, for an outside observer, a clock approaching the trajectory of a photon would appear not to tick. From a different point of view, we can reason by analogy to Euclidean geometry. When we have a curve through space we can talk about the tangent vector at any point. But the length of the tangent vector is not a well-defined concept. Only its direction is. You can define the length of the tangent vector for a particular parameterisation p(t) but for the curve itself, i.e. the set {p(t) : t in R}, the length of the tangent vector is not defined. It is similar for a curve through spacetime. We can parameterise a curve with some variable x and write the curve as {p(x) : x in R} but this is just a set and the length of the tangent vector of p(x) depends on which parameterisation you chose for the curve. But the data of how a particle moves through spacetime is only the curve {p(x) : x in R} itself, not the function p(x): different parameterisations describe the same physical situation as long as the set {p(x) : x in R} is the same.