The force got distributed in the same way. You just chose two extreme examples that look like they are different.
When you pull the rope towards you, you increase the tension on the part of the rope nearest you. Now, to be completely accurate we would have to consider how fast this change in tension is distributed. It gets distributed at the speed of sound, which is really fast. Faster than anything in your experiments. So we'll ignore it... but the effect is indeed there.
So when you pull on the rope you increase the tension force. Now that the tension force is higher than the force of gravity pulling on the rope, the rope actually gets pulled upwards. You can sometimes see this at the bottom of the rope. The key here is that you build up a gravitational potential. The system wants to let the rope extend back down fully, so that it is at the lowest height possible.
So now you have two forces. You have the force of tension from your hand, which pulls the top of the rope towards your body, and pulls the rest of the rope along the length of the rope, and the force of gravity, pulling the rope down. Intuitively, the force of tension on the rope is going to cause the rope to curve. You're pulling it horizontally at the top of the rope, but the rope can only pull along its length so the force is all vertical at the bottom.
Gravity is going to be trying to undo this curve. The curve means the rope is slightly higher on average than it could be without a curve. So, over time, gravity is going to straighten the rope.
What you have here is a balance. If you move quickly, you apply a lot of force with your hand, and its effects dominate. The result is that the rope sort-of follows your hand, being pulled along a smooth curve. You apply this force quickly, so there is not much time for gravity to act. On the other hand, if you move slowly, with less hand force over a longer period of time, gravity has more time to act. Thus the rope ends up straighter.
The rope is never straight, no matter how slowly you move your hand. There's always some curvature caused by this balance. However, you likely cannot see it with the naked eye. Its effects are dwarfed by all sorts of other effects, like the fact that your hand is never moving perfectly horizontally. It bobs up and down as a natural effect of how our brain controls our muscles. But the curving is there. It's not like there's a sudden point where the rope goes from moving as a straight rod to moving on a curve.
Incidentally, even if the rope was replaced with a straight rod, this effect would occur. However, rods tend to be stiffer, so they deflect a lot less.