Your definition of temperature is essentially correct. If you came upon this definition on your own, I congratulate you, because it is much deeper and much closer to the fundamental statistical definition of temperature than the usual superficial statements about average kinetic energy that you will usually hear.
The following statement, which I'm paraphrasing from you, is a reasonably good definition of temperature:
Temperature is the property of an object that is the same for two objects that are in contact with each other if those two objects maintain no net exchange of energy between them.
This definition is a bit indirect. It doesn't tell you how to actually calculate an object's temperature, for example. But it's an accurate definition.
Now what about the issue with the video? (By the way, you haven't shared a link to the video so I have to go based on your description. If you share a link and it turns out I misunderstood, I'll edit/delete this answer.) The ice is colder than the metal/plastic, so in both cases energy is transferred to the ice. But you say that one transfers the energy faster than the other. This is not actually a problem for your understanding! It sounds like you're thinking, "Metal transfers energy faster than plastic even though they are at the same temperature, so the net flow between them would still be non-zero." (Let me know if I got you wrong.) But that's not quite right.
The only issue here is that you're comparing different pairings of the objects. The rate at which a substance transfers energy is not a universal property of that material. It depends on the situation and will change throughout the process of coming into equilibrium. The metal and plastic both at the same temperature do transfer energy to each other at the same rate if they are in contact. But metal at a warm temperature transfers energy to ice at a cold temperature at a different rate than plastic does to ice at the same two temperatures. Nevertheless
as the metal transfers more and more energy to the ice (or water if it melts), it will become progressively less willing to give more energy, and the ice will try progressively harder to give energy back to the metal until they reach a point when they are both equally willing to give/take energy. Then the net flow will become zero, the situation will reach equilibrium, and we would say they have the same temperature. The same will happen for the plastic and ice, although the details of how willing the plastic is to give energy at the beginning may be different, so that the melting begins more slowly. But in the end the plastic and ice/water will again "negotiate an agreement" and reach net zero flow.
This way of thinking about temperature is related to a concept called entropy, which very roughly is (the log of) the number of different microscopic configurations a system can be in with a given amount of energy. One of the principles of statistical mechanics is that systems like to maximize the number of different microscopic configurations that are accessible to them. It's usually the case that as an object's internal energy increases, the number configurations it can access increases. In your metal/ice scenario, initially the ice could dramatically increase the number of possible configurations if it had a little more energy. This in fact is what it means to be cold. Meanwhile, the metal's set of configurations would only decrease a relatively small amount if it lost energy. This again is what it means to be warm. So, fine, the metal gives a little energy to the ice, the ice opens up a lot more configurations, and the metal only loses a little. But as the metal gives more and more energy, it starts to lose more and more possible configurations for each chunck of energy it gives up. Eventually, it is no longer willing to do so, when the "benefit" to the ice equals the "harm" to the metal. The plastic and ice do the same thing, but the plastic, due to the details of its internal structure, may be more prone to losing configurations as it gives away energy, so it was less willing to do so from the very beginning, and the melting proceeded more slowly.
How does this all connect to the common statement about kinetic energy? If the particles that a system is made of can move around in some way, then as a system gains energy, the particles can move around more, roughly speaking, and that typically translates to more possible microscopic configurations. But you see that the explanation given above is much more fundamental, because it emerges from just considering any kind of configurations that the system have, which may not just be motion. For example, the Ising model is a model of magnetic materials that has all the properties such as temperature and phase transitions that you are familiar with, but nothing to do with kinetic energy. Your definition of temperature covers systems like this equally well.
If you're still not tired of reading all this, you can learn more about these topics from Schroeder's very nice book, which is also apparently summarized in this video.