So, I saw a video on YouTube that says the banana we eat today is not what a banana looked like years ago. Since the banana has been genetically modified over the years, does that qualify as evolution?
Yes. Modern domesticated bananas have evolved from ancestral wild species under the influence of humans. Quoting D'Hont et al.: (1)
The [banana] domestication process started some 7,000 years ago in Southeast Asia. It involved hybridizations between diverse species and subspecies, fostered by human migrations, and selection of diploid and triploid seedless, parthenocarpic hybrids thereafter widely dispersed by vegetative propagation.
Modern bananas have undergone extensive evolution from the original wild species. The selection pressure has in part been provided by humans, who have selected the most nutritious/tasty/desirable examplars and facilitated their reproduction. This can be considered "artificial selection" as opposed to "natural selection", but does not change the fact that an evolutionary process has taken place. The UC Berkeley website "Understanding Evolution" gives the following definition of "artificial selection".(2)
A process in which humans consciously select for or against particular features in organisms. For example, the human may allow only organisms with the desired feature to reproduce or may provide more resources to the organisms with the desired feature. This process causes evolutionary change in the organism and is analogous to natural selection, only with humans, not nature, doing the selecting.
Thus, whether the selection is considered "natural" or "artificial", the end result is evolution.
You could say it clasiffies as evolution only if you accept that the artificial things humans have done to bananas are actually natural occurences. I personally don't think they are, because intent is at play here, and evolution is a natural process without intent.
Evolution is not when something gets better. That's the misconception behind your question.
If humans would just happen to eat the tasty bananas first, what would happen is negative selection and all the remaining bananas (which lived long enough to reproduce) wouldn't be tasty. Bananas would get less tasty in order to survive.
What actually happened goes against the natural process of selection, and is something completely different to the concepts of evolution which led to us evolving intelligence, for example. This is just selective breeding. Nothing as grand as evolution.
Evolution is a natural process which led to intelligent species from single-cell organisms, and is a continuous process of random mutations being selected for or against by natural causes. Evolution has no intent and is a natural random process. A grand one, as all biology depends on it, and we depend on it.
Genetic engineering, on the other hand, has intent to improve things, and happens on a much smaller scale than the natural process of evolution. Genetic engineering of any kind, whether it be tinkering with the genes or just inbreeding to create a new breed, is artificial and is being done by an intelligent creature as opposed to nature. Calling genetic engineering a tool for evolution is the same mistake that eugenics proponents made.