I understand that in order for an object to maintain circular motion, its velocity vector must be travelling perpendicular to its position vector and constantly changing inwards, hence an acceleration towards the center of the circle. I know that the acceleration towards the center is typically caused by other forces, like tension on a string, and that these are called centripetal forces I believe? However, objects in circular motion tend to want to be away from the center instead of towards. A bucket of water tied to a string and twirled around in a circle will result in the water staying in the bucket: if the water is exhibiting circular motion, would it not thusly be accelerating inward, and thus escaping the bucket? I've heard that it's a difference of frame of reference, but even looking from out to in, I can't see how the water would be accelerating inward and yet remain in the bucket without support. Would there not be some force pushing the water into the bucket? And yet, centrifugal force is considered a fictitious force. I don't understand. I know I understand some level of physics but please explain it like I'm 5 because I can't seem to actually understand this.
Your third point clarifies some things for me a lot. I seem to have forgotten that acceleration describes a change in direction and/or a change in magnitude of the velocity vector: I recall now in my physics textbooks that objects in non-constant circular motion have a tangential acceleration, and the total acceleration lies somewhere between those, but if the velocity remains constant, then the only acceleration is the centripetal acceleration describing the change in the velocity's direction.
However, I still have some questions about points one and two. I understand that things in circular motion want to fly out tangentially away from the center, not radially away. Yet, in so far as I can observe, objects do seem to press outwards radially. In the example with the bucket of water, the water sticks to the bottom of the bucket instead of pressing against the side wall. In another example, that of those carnival rides that spin people around in a saucer (gravitron I think it's called?), the carmival goers tend to stick to the wall of the ride as though they were being flung out radially, instead of rolling along the edge or something else. I guess it's this disconnect between what I know is correct (objects fly tangentially to their circular paths) and what I observe (objects stick to the wall radially away from the center).