This is because the first postulate of special relativity says speed of light is constant, and so in order to compensate for the extra horizontal distance covered by the moving clock, its time must be increased...
That is partially incorrect in SR. There is no extra distance covered by a moving object, as surrounding space is viewed within its own frame of reference but to the contrary, less travelling distance is covered due to Lorentz space length contraction therefore this is combined with a slower rate of time thus time dilation, as perceived by an external inertial observer so that the speed of light $c$ remains the same for all observers independent frame of reference. As space shrinks in the direction of motion for a constant speed moving object or observer, time must expand for the external inertial observer to keep light speed $c$ constant. This is due to the transverse propagation of light (information) coming from a longitudinal moving object. Space increases for light the faster the moving object, which shrinks more space in its line of motion and increases the curvature of spacetime. Therefore an external inertial observer will see the object travelling for a longer time reaching its destination.
The above is for constant speed Special Relativity theory SR.
For acceleration time dilation equivalent to gravitational acceleration time dilation effect, of General Relativity theory GR, the effect is similar to constant speed SR time dilation and an additional time dilation effect, only now the accelerating observer will not see the inertial observer time dilated (i.e. also assuming inertial observer is not inside a gravitational wheel) due to GR time dilation but only due to SR time dilation which is always present regardless of if relative moving objects are also accelerating or not.
So, answering your question, in SR time dilation the relation of the inertial and moving observers regarding time dilation effect is reciprocal whereas in GR time dilation, this is not true.