In reality, the phrase "a moving clock runs slow" does not necessarily mean "a moving clock is seen visually to run slow." A clock moving directly away from an observer appears visually to run slow, but a clock moving directly towards an observer appears visually to run fast. In both cases, what is seen visually is given by the Doppler expression, which is always different than the time dilation expression. In both cases, the time dilation expression, used appropriately, does apply.
Consider the following example.
Assume that Alice is moving with constant speed directly towards Ted. When Ted uses his telescope to watch Alice's wristwatch, he sees her watch running at a faster rate than his watch. Ted sees Alice's moving watch running fast, not slow! Ted sees this because of the Doppler shift. Because Alice moves towards Ted, the light that Ted sees from her watch is Doppler-shifted to a higher frequency. But the rate at which a clock or watch runs is like frequency, i.e., a second-hand revolves at a certain frequency, and all frequencies are Doppler-Shifted., so ted see Alice's wristwatch running fast.
To explain what "A moving clock runs slow." means, I first have to explain how Ted (with help from Bob) establishes his frame of reference.
Starting from Ted, a series of metre sticks, all at rest with respect to Ted, are laid end-to-end by Bob along the straight line joining Alice and Ted. At each joint between two consecutive metre sticks, Bob places a small clock. The metre sticks and clocks all are at rest with respect to Ted. Initially, none of the clocks are running; before turning them on, the clocks have to be synchronized. To do this, Ted directs a laser pointer along the line joining Ted and Alice, and then sends a flash of light. Each clock is turned on when the flash of light reaches it. The speed of light is not infinite, so the time taken for the light to travel from Ted to each clock has to be taken into account. To do this, the clocks' hands are set initially as follows. The clock one metre away from Ted is set to the time taken for light to travel one metre; the clock two metres away from the tower is set to the time taken for light to travel two metres; ... .
This whole setup of metre sticks and clocks establishes Ted's reference frame.
Now, As Alice moves toward Ted, Ted uses his telescope to watch Alice's wristwatch, and to watch his clocks. First, he watches one of the distant clocks in his reference frame. The time he sees on the clock is the time at which the light he sees set out from the clock, so Ted sees an earlier time on the distant clock than he sees on his wristwatch. Because the clock is stationary in his frame, Ted does, however, see the distant clock running at the same rate as his watch. Similarly, Ted's sees all the clocks in his frame running at the same rate as his watch.
As Alice approaches Ted, she whizzes by clock after clock of Ted's reference frame. Using his telescope, Ted sees that Alice is beside a particular clock, and he notes the time on her watch and the time on the clock adjacent to her. Some time later, Ted sees Alice beside a different clock, and he again notes the time on her watch and the time on the clock adjacent to her.
Ted checks his notes, and he finds that the time that elapsed on Alice's watch as she moved between these two clocks of his frame is less than the difference of the readings of the two clocks. This what is meant by "A moving clock runs slow."
Unfortunately, "time dilation" in general relativity and "time dilation" in special relativity often have different operational meanings. Suppose observer A hovers at a large distance from a Schwarzschild black hole, and that observer B hovers near the event horizon. If observer A uses a telescope to observe B's watch, A will see B's watch running more slowly than his own watch. In this context, gravitational time dilation is something that is seen visually.