Yes.
However it is much better to recognize events as locations in a 4d spacetime and that a clock ticks based on how distance is measured on this 4d spacetime between two events.
So you being born is one event. And your first birthday is another event and you can imagine the xy plane as showing where you are and time being the z axis showing when it happened and then as you go around the sun you make a helical path.
Then the rate your eyes and brains and stomach and watch work is based on needing a certain length along this curve instead of a certain amount of time. Thus someone that runs around you is like a corkscrew around your helix and they could age differently even if they were with you for those two birthdays.
Then the two time dilations become the same. You have to learn how to measure distances in 4d the correct way. And it is different than you are used to for two reasons. Firstly, it has a Minkowski geometry not a Euclidean one. And secondly it can be curved.
Not just a curved space but a curved 4d spacetime.
And since you have to learn both these things one is what we teach by relative motion (Minkowski geometry) and one is what we teach by gravitational time dilation (curved spacetime). You need to learn the first before you are ready for the second and when you use the second you always also have to use the first.
You are saying clock ticks are the unit of measurement in spacetime. Would I be correct in saying a moving object would travel at a certain tick rate while an object in [gravitationally] compressed spacetime would travel at a higher tick rate?
When you measure you measure from two ends. Pick different ends and you get a different measurement. Pick a different path between them and you get a different measurement. A geodesic is the most extreme measurement, for instance airplanes travel on geodesics of the curved earth surface because it saves fuel to take a path with a shorter measurement.
A similar thing happens in spacetime. The reason light deflects around a star is that the path with the most extreme measurement isn't the one where you start out pointing straight at your destination. So when you are pointing straight at your destination you are optimally pointed for going somewhere else so you go somewhere else. This is a bit confusing since this just changes what it means to be pointed someplace. But for instance you might end up going in a circle if the most extreme way to get to the opposite side is to start out going in the direction you are going. So of all the ways to fire rockets and rush to the other side one of them has you age the most. And that motion is the way you go if you don't fire any rockets but are aimed the right away.
So you can go in many paths if you fire your thrusters and use rockets. But there is one path that has more time measured along it than any other lath with the same ends. And that one starts out heading a certain way. If you start out heading that way and don't fire any rockets that is the path you will take.
That's how things move. Rockets deviate you from that optimal path, and that is why you age less when you use your rockets. But you will naturally (with no rockets) move into a gravity well if that helps you age more on your path. You always naturally move in the way that ages you the most out of all the paths that go places you go.
It sounds like you are saying an object in freefall(optimal path) in a gravity well would age at the same rate as an object in freefall in free space (contingent upon them both traveling in the same direction and rate of speed). The act of resisting the natural path is what causes gravitational time dilation. Can this be correct?
No. If you have different ends you have different paths. So free space and a gravity well have nothing to do with each other.
Pick some ends. Step one you measure from the same ends. Step two there is a natural path that ages the most between those ends. That is the path you take between those ends. When you are pointing some direction that direction is the direction that is optimal for some end. That is the end you go to.
For instance if there is a planet going around a star. For instance the earth at summer is heading to the earth in fall. It might look like the direct path is not a circle, but such a path takes you near the sun where you age less so the path that ages you the most to get there is the circular path those you get there by moving tangent to the sun. If you pointed yourself that other way you would move on an ellipse that has you spend lots of time much farther from the sun to compensate for the time you spent near the sun.
