Clarification about what makes a system isolated

I am a grade 12 physics student and I just need some clarification about what makes a system isolated. I've read the definitions online, but they still don't make a lot of sense. For example:

When people jump on a rotating merry-go-round, the angular momentum decreases. However, according to my physics teacher, when people jump off from a spinning merry-go-round, conservation of momentum does not hold because it is not an isolated system. Why is one case isolated and the other isn't?

Also, if an object suspended to the ceiling by a cord gets hit by an object (ex. a bullet), why is the moment of impact the only time the system is isolated, and not when the object+bullet moves towards the ceiling?

A system is isolated when no important external forces act on it.

You need to be clear on what is the system and what are important external forces.

When people jump on a rotating merry-go-round, the angular momentum decreases.

When a person jumps onto a merry-go-round, this is an inelastic collision. The person and merry-go-round collide and stick together. During the collision, the person and merry-go-round exert forces on each other and accelerate until they reach the same speed.

If we choose the system to be the merry-go-round. The person is not part of the system. The force on the merry-go-round from the person is an external force. This force slows the merry-go-round. Angular momentum of the merry-go-round is not conserved.

We could choose the system to be the merry-go-round plus the person. In that case, the forces the person and merry-go-round exert on each other are internal forces. The only external forces are gravity and the reaction force from the ground.

We will ask if the angular momentum changes between 1) the person flying through the air and 2) the person landed and holding onto the merry-go-round.

We will first only consider rotation in a horizontal plane around a vertical axis. Gravity on the person is vertical. It does not change his horizontal motion. Gravity + reaction holds the axis of the merry-go-round still, but has no effect on the rotation of the merry-go-round. For this much, the external forces do not affect the part of the motion we are interested in. They are unimportant. We can conclude the angular momentum of the system is conserved.

We have cheated because a person flying through the air has a vertical component of velocity. So the system isn't entirely rotating around a vertical axis. When the person lands, the merry-go-round stops his vertical motion because reaction forces from the ground prevent it from moving in that direction. So reaction force does affect that part of the problem. The system is not isolated, and angular momentum is not conserved if we consider everything.

However, according to my physics teacher, when people jump off from a spinning merry-go-round, conservation of momentum does not hold because it is not an isolated system.

I expect here he is considering changes to the person/merry-go-round system between 1) the person holding on and 2) the person landed and stopped on the ground.

Friction from the ground has changed the person's horizontal velocity. This does affect the motion we are interested in.

why is the moment of impact the only time the system is isolated, and not when the object+bullet moves towards the ceiling?

Tension from from the cord and gravity are the only external forces. The bullet moves very fast horizontally and not so much vertically. We can ignore the vertical velocity of the bullet.

Just after impact, the bullet/object system is moving horizontally. External forces don't affect it.

A bit later, the cord is at an angle. The external forces now oppose the horizontal motion and decelerate the system.

Also the tension in the cord has changed. Before, it was just exactly enough to prevent the cord from stretching. This kept the bullet/object from accelerating downward. Now it is just exactly enough to keep the cord from stretching. The bullet/object is accelerated upward.

Both of these changes make the external forces important, and the system not isolated.

It all depends on the system under consideration (that's fancy text for the system you're thinking of). In the first part, you could argue that when people jump off, the momentum of the entire Earth (which is what you landed on) is conserved because it's an isolated system. Your physics teacher is assuming that you aren't treating the entire Earth as part of the problem. This is actually rather sensible, since the mass of the Earth is $10^{24}$kg, while people are only ~$100$ kg. The momentum imparted to the Earth is way negligible then!

I don't understand the second part very well. Naively, the object + bullet should be isolated, simply because there's nothing else in the problem.