Understanding two instances of different time dilation I needed clarification as to two aspects of time dilation which seem distinct for me.
Consider an observer inside an accelerated rocket and another observer outside of it. The outside observer will perceive the time for the observer in the moving rocket as running slower, because of time dilation due to relative motion.
However, inside the ship, there will also be time dilation because of its acceleration (as it would happen in a gravitational field). However, this second aspect of time dilation will only happen inside the accelerated environment, and it won't affect outside observers (because they are not subjected to the same acceleration). Is that generally correct?
As explained here by Feynman, clocks at the bottom of the spaceship will run slower than clocks at the top. However, this particular time dilation doesn't affect observers outside of the ship. Outside observers might perceive both clocks running differently, but it doesn't affect their own. Is that correct?
 A: We can separate the acceleration effect from the effect of relative velocity.
Suppose a ship with uniform acceleration. It is always possible to imagine an inertial observer that is momentarily with the same velocity of the ship. We say it is a momentarily comoving frame. It can be for example someone that jumps from it to the outer space. Comparing the clocks in this moment, both agree that time is running slower in the ship.
But we can also imagine that the ship passes by an inertial observer with a difference of relative velocities. In this case, for the inertial frame, time is running slower in the ship. Because even without acceleration it is the expected outcome. And the acceleration effect is also to slow time.
But for the ship, it depends on the relative velocity and the magnitude of its acceleration. For small relative velocity, time is slow in the ship. For high relative velocity, time is faster in the ship.
A: There are no fundamental rules about "time dilation" in special or general relativity. There are just rules about geometric relationships in spacetime, and proper times of objects.
For example, "gravitational time dilation" is just the name for a spacetime version of the well-known geometric fact that if you have curves bending around a common center, the curve farther from the center will be longer. The starting positions of runners on oval tracks are staggered for that reason. Any formula you may see for gravitational time dilation just quantifies that geometric fact in a particular situation. There's no extra gravitational time dilation effect on top of the geometric relationships.
So this:

However, inside the ship, there will also be time dilation because of its acceleration (as it would happen in a gravitational field)

isn't really correct. Clocks define the passage of time. A clock can't run slower than itself, and there's nothing else to compare it to here, so there's no meaningful concept of time dilation in this setup.
On the other hand, if you have two clocks at the top and bottom of an accelerating ship, they will measure out different spacetime lengths (different elapsed times) for the abovementioned reason; that's gravitational time dilation.

However, this particular time dilation doesn't affect observers outside of the ship. Outside observers might perceive both clocks running differently, but it doesn't affect their own. Is that correct?

As I said, time dilation doesn't really affect anything. It's just a set of facts about geometry. An accelerating rocket ship emitting light pulses that are seen by someone moving inertially is a fairly complicated setup and you may not find a ready-made "time dilation" formula that applies to it. But you can always work out when the light pulses will arrive by general techniques.
