I understand this from the explanation, but doesn't this conflict with
the concept that time seems to run slower for objects moving relative
to an observer? This explanation would lead one to believe that the
rate at which an observer sees a moving object travel through time
depends on whether the object is moving towards or away from the
observer.
I think it is worth to mention , that Relativistic Doppler Effect itself says nothing about clock rate, although includes time dilation. That depends on chosen reference frame whether “time seems to run slower for object moving relative to an observer” of “time runs faster” or even at the “same rate”.
To demonstrate that we can consider Relativistic Doppler Effect in different frames of reference.
Let’s consider that case, when a source of radiation approaches an observer with velocity close to $c$. In this case the observer will see a blueshift of frequency of “infinite intensity”, as Albert Einstein justly noted in 1905 article:
https://www.fourmilab.ch/etexts/einstein/specrel/www/ - § 7. Theory of Doppler's Principle and of Aberration
Measured Doppler blueshift of frequency is absolute effect; it cannot depend on chosen reference frame. However, relative contributions of time dilation are frame - dependent.
Let’s assume that an observer considers himself being “at rest” and a source of radiation approaches him.
„Classic” Doppler effect in this case will be:
$$f_0= \frac{f_s}{1-\cos\theta_s\cdot v/c}$$
Hence, in ”classic” case, if $v=c$ all wavefronts will gather straight in the front of the source into one and would hit the observer at once, like fighter jet’s sonic boom.
However, there is dilation of the source’s clock. The source oscillates $1/\sqrt {1-v^2/c^2}$ times slower, so relativistic formula looks like that:
$$f_0=f_s \frac {\sqrt {1-v^2/c^2}}{1-\cos\theta_s\cdot v/c}$$
Hence, dilation of source’s clock “slows down” oscillations. Measured frequency will tend to infinity as velocity of the source approaches that of light, but at “slower” rate than classic one.
On the other hand, the observer may say, that he was in motion himself in the reference frame of the source, so his own clock slowed down.
In classic non - relativistic case, if an observer moves towards a source, measured Doppler shift will be:
$$f_0=(1+\cos\theta_s\cdot v/c)f_s$$
According to this formula, if an observer approaches the source with velocity $v=c$, maximum measured frequency $f_o$ would be equal to $2f_s$, since wavefrons and the observer approach each other with equal velocities.
Since the observer’s clock slows down, now we must divide this frequency by $\sqrt {1-v^2/c^2}$. That means, due to time dilation, the observer turns into a “dawdler”. While he lives just one minute, the world around him may live one year. So, he sees “usual” processes around him as very fast, as if someone plays movie in fast forward mode.
$$f_0= \frac {(1+\cos\theta_s\cdot v/c)}{\sqrt {1-v^2/c^2}}f_s$$
We see, that frequency $2f_s$ turns into infinitely intense, as velocity of observer approaches $c$.
Though these formulas (for moving source and moving observer) at first glance look different, actually they are the same.
http://www.feynmanlectures.caltech.edu/I_34.html - 34.6 - The Doppler Effect.
This way an observer can interpret Relativistic Doppler Effect as he wishes. He can say anything he wishes about a rate of a clock that approaches him. If he considers himself being “at rest”, he will say that clock is ticking slower. If he considers himself being “in motion”, he will say that the clock is ticking faster. If he thinks, that he himself and the source approach each other with equal velocities, he will say, that source’s clock is ticking at the same rate as his own.
http://www.mathpages.com/home/kmath587/kmath587.htm
“Since both emitter and receiver have the speed v relative to this system of reference, there is no differential time dilation”
Quite interesting, that Albert Einstein considers Relativistic Doppler Effect in the reference frame of “stationary” source. His formula shows, that frequency $2\nu’$ not decreases, but increases $1/\sqrt{1-v^2/c^2}$ times, i.e. moving observer "sees" or better to say "interprets" that the clock “at rest” is ticking not slower, but faster.
The same thing happens if an observer measures time dilation by means of synchronized clocks. If an observer employs Einstein synchrony convention, he will measure, that moving clock is ticking slower.
However, if an observer considers as moving himself in the reference frame of the source, he must synchronize clocks in his frame according to time of source’s reference frame. This synchronization will be equivalent to Reichenbach synchrony condition, which keeps two – way speed of light isotropic while one – way speeds of light anisotropic. In this case this observer will see, that a clock “at rest” is ticking not slower, but faster, exactly as in the case with Doppler Effect above.
https://en.wikipedia.org/wiki/One-way_speed_of_light
