Make Extremely Specific Classifications, and have a test for all people who reach a certain age. Anyone who exhibits traits that are beyond the government's strict definition of "normal" is thereby "super"
World Governments and Organizations can solve the definition problem by being extremely specific with what constitutes "Super" or not. The best way to classify someone is to make a test for certain traits. Anyone that exceeds what the government considers normal is therefore classified as superhuman.
Tests could include basic things like Speed, Strength, and Intelligence tests as a baseline.
If you can lift more than X-amount of weight, then you are far outside the scope of human strength and are therefore super.
If you can run more than X kilometers in T seconds, then you are considered to have super speed.
An intelligence test is a much harder and more abstract, but if you are able to memorize thousands of images in a single sitting, calculate and multiply thousands of digits in your head without a calculator, or even more impressive feats, then you would be considered super smart.
Other anomalous abilities would be harder to test for, so the government would have to be creative and very thorough to avoid missing out on certain people.
To test for flight, you might ask a person to jump a certain distance in the air. To test for telepathy, ask them to read a person's mind and guess what cards they are holding from another room. To test for laser vision, ask them to stare at a piece of paper as intensely as possible. So on and so forth.
Obviously, you can't catch every power in a single go. If there's a wide variety of abilities you're always going to find one that's either incredibly rare or wholly unique. That's why I would suggest the governments should not only test for abilities but do so regularly and with great specificity.
Rather than classifying a human as "having no ability" it would be better to classify them as "having no ability that we know of yet". To the best of the researcher's knowledge, they could not find anything out of the ordinary with the person, but it might be something they have never thought of.
Obviously, there are going to be cases where there are false positives or false negatives, so the point of repeated tests is to make sure that the results are as accurate as possible, and the more scientists learn about the powers, the less likely they are to make a mistake.
Sometimes an ability will be so rare or unusual that it will fall outside the previous guidelines, or it is so specific that it's nearly impossible to scan for. For example, what if an ability only comes out when a person is in a life or death situation. It would be unethical to risk someone's life to test them for that, so what if you show them a jumpscare from a movie or something like that? A slightly more extreme idea would be to give people an adrenaline shot to trick their bodies into thinking they are in danger, forcing any latent ability to activate if it is indeed activated by fear.
That would be an example of a false negative. A person is super, but they didn't show anything on the test. Legally, they are not super because they have no ability on record. While they might have something hidden, the government has decided they have nothing unusual about them.
A false positive would be a person who is either "peak human" or someone who passed the test either through strange happenstance or purposeful rigging.
By peak human, I mean a person who manages to, by sheer force of will and training, pass all the tests that they are given. They passed the strength and speed test through a near Olympic level of training, and they have a genius-level intellect they have honed through study and effort. They may never be able to achieve the feats of people who have superpowers, but they are immensely impressive in their own right.
The test could also go wrong either through accidents or manipulation of the results. A family of superheroes might not want to admit that their precious child has no abilities, so they bribe all the judges to have them right down that their child is in fact super. Or, a person could pay a superhuman to either pretend to be them for a day or act from the sidelines, either feeding them the results they need through an earpiece or using their powers to intervene. A telekinetic, for example, could make their friend pretend to fly for a while.
No system is perfect, and there are definitely going to be super individuals who slip past the notice of the system. There are also people who might be able to pass through sheer luck, remarkable strength, and talent, or by playing the system. Such people would be considered "super" even though they are not in actuality. Money could very well be a superpower if a judge can be bribed with some crisp hundred dollar bills.
Honestly, I love the idea of a person being able to earn the title of "super" through being an Olympic level athlete, a math prodigy, or a remarkably skilled musician, or any other vast talent that they could have.
So, what classifies a person as super? Whatever society or the government says it is according to their guidelines.
The idea and classification might be different between countries or even cities. It also might be suggestible to have levels such as S, A, B, C, and so on.
C is the lowest level. It is also referred to as the "common" level. You might have a power but, as far as anyone is concerned you are considered indistinguishable from a normal human being. Humans are lumped into this category too.
B is the next highest. If you're an Olympic level athlete or math genius, you might be able to get here, but you'd have to be incredibly talented.
A is even higher than that. The things they can accomplish are far beyond anything a normal person could be able to do.
S is even farther up the ladder, seeming mostly godlike to people of lower levels.
There might even be a world ranking system
Every person in the world could be given a certain "score" based on their tests, used to determine what their overall ranking as a superhuman is.
The person with the highest score is therefore on record as the most powerful person on the planet. This score is both an honor and a burden, because it means that person is arguably the most dangerous person alive.