I see. Good point, it depends on how you define four velocity and how exactly you interpret "moving through time with the speed of light". I personally think that "we move through time at the speed of light" is a very confusing way to explain that physical fact about how clocks tick. I'd explain the same fact as follows:

If you have a clock moving through spacetime along some path p(r), put tickmarks at regular intervals [1] of its arc length ∫|p'(r)|dr. Those tickmarks indicate when the clock ticks.

[1] e.g. choosing the length of an interval by matching it to a one second tick of a stationary reference clock.