Whether accelerating or inertial, any moving object will traverse a continuum of spatio-temporal points in it's path, correct?
But if an object is sitting at rest (rest frame), then the object is not in transition between spatial coordinates. However, the object does transition from one moment to another (however you define "time" is arbitrary for this discussion), therefore the object traverses a continuum of "time-like" points. So although there is no motion through space, there is still a perceived transitional displacement through time (moment to moment). Since rate of spatial displacement (measured by ruler) as compared to time displacement (measured by clock) is definition of "speed", then it stands that 0 speed is equivalent to 0 spatial displacement (in appropriate rest frame). This does not mean that the demoninator has gone to zero, only that the numerator has gone to zero (numerator is spatial displacement, denominator is temporal). There is still a temporal displacement, so we call this: "displacement through time", although "speed of time" may or may not be defined (which it is not, according to the above definition for speed).
Temporal displacement is same thing as "temporal motion" (displacement = motion), which is motion through time.
As rate of spatial displacement per unit time approaches 'c' for a space bound rocket (as measured by the rest frame of, say, Earth), motion through time for that rocket will approach a stand still (once again, as measured by Earth observer). The defining of an appropriate reference frame here is critical, otherwise "motion" has no specific meaning at all. The notion of "proper time" of different frames bears no significance to this discussion; we are speaking of measurements as taken by one frame, and one frame alone (only proper-time of Earth need be mentioned).