The gravity one is correct; the speed one is incorrect.
The basic reason that the speed one is incorrect is: suppose that Alice is moving very fast relative to Bob. You want to say that Alice's clock is moving slowly, because that is how Bob sees it. Unfortunately for you, Alice also sees Bob's clock moving slowly.
Relativity is all about this strange observation: when we accelerate, stationary clocks in front of us appear to tick slightly faster, and clocks behind us appear to tick slightly slower, in proportion both to the magnitude of the acceleration and the distance that they are away from us. Everything else -- length contraction, time dilation -- can be derived from this with some intermediate matrix mathematics. The exact formula can be written this way: when you stop accelerating and keep at a constant velocity $v\ll c$, then your times $t'$ will relate to the stationary times $t$ by $t' \approx t - v x/c^2$: stationary clocks which used to appear in-sync are now desynchronized. Note that you can now scatter some clocks which you think are synchronized and the stationary frame will think that those are desynchronized, $t \approx t' + vc/c^2.$ Once you stop accelerating you both have equivalent perspectives related by mathematical formulas.
Going faster and faster one exponentiates a certain matrix and derives $t' = (t - vx/c^2)/\sqrt{1 - (v/c)^2}$ implying that the one frame sees your frame ticking slow; but the inverse to this is actually $t = (t' - vx'/c^2)/\sqrt{1-(v/c)^2}$ for your coordinate $x'\ne x$, and therefore you see their clocks ticking slow, too. It's a very symmetric relationship and the effects are only seen one way or the other depending on how you accelerate in order to meet again.
Let me explain the twin paradox in this way. One twin, Alice, accelerates in the vicinity of Earth, then travels at constant speed out to the stars, then accelerates twice as hard towards Earth to turn her ship around, then traverses the space again, then decelerates back to rest at Earth. Her twin, Bob, stays on Earth and can imagine that he sees Alice's clock moving in slow motion for the entire journey due to the $1/\sqrt{1 - (v/c)^2}$ factor. As we've said, Alice sees Bob's clock move slowly as well: however in this entire sharp acceleration at the other star, she is accelerating towards Earth and sees Earth's clocks tick faster, in proportion both to her acceleration and this distance that Earth is away from her. That effect overwhelms her sense that Bob's clock has moved slowly, and it overwhelms it considerably because of this very long distance, which is negligible at the beginning and end. Even though Bob's clock moves slowly the rest of the journey back, they both agree on the aggregate effect, that Bob's clock has ticked faster overall and Alice's has ticked slower. What's objective is really the accelerations involved, Bob has not accelerated while Alice has.
If you understand the twin paradox now, you have everything you need to understand gravity.
When you are stationary in a gravitational field you are secretly accelerating upwards; physics is meant to be done in free-falling reference frames and yours is not free-falling. Consequently you see a more-permanent fast-ticking of clocks that are above you, and they see a more-permanent slow-ticking of your clocks. You see, gravity gives you the option of accelerating in order to remain in-place, and that changes everything. As you pass near the black hole you accelerate away from it simply to avoid falling in; but as you accelerate away from it the world outside the black hole seems to be ticking very fast relative to you.
