There are two commonly accepted definitions of probability:
- Frequentist probability: The probability of X happening is the proportion of times that X actually does happen, out of all times when X could have happened. This is an objective fact about the universe, and can't be argued with on the basis of opinion. However, our measurements of this value may have some uncertainty. This uncertainty can (in theory) also be quantified objectively, but this usually requires us to make some assumptions about our measuring process (for example, we may need to assume that our sampling method is unbiased, and it's often hard to prove that this is actually the case).
- Bayesian probability: The probability of X happening is a measurement of how certain we are of X happening, based on the information available to us. Since different information will be available to different people, this is inherently subjective. However, this does not mean that you can just make up whatever probabilities you like. Your probabilities must be consistent with the evidence and with the Kolmogorov axioms, which also entails that you must use Bayes' theorem to update your probabilities when new information becomes available. Since frequentist probability also obeys those same axioms, all theorems of one kind of probability are also true of the other, but have a different real-world meaning because of the different definitions of "probability."
(Some people think that one of these definitions is the only correct definition. But the term is used with both meanings, so it would be pointless to crown one of those groups "right." We would just have to invent a new word for the other concept anyway.)
Some probabilities may only make sense in one interpretation or the other. For example, "what is the probability that candidate X wins the election?" is obviously asking for a Bayesian probability, as this exact election (with this exact electorate and slate of candidates) will most likely never be run again. On the other hand, questions of statistical significance usually revolve around a slightly convoluted use of frequentist probability (in simple terms, you take a statistical measurement, assume that it was just a fluke and then ask "if this measurement is just a fluke, then what was the probability of getting a measurement at least this extreme?" - if that conditional probability is low enough, then you can reject the "it was just a fluke" explanation as implausible).
To focus on the question you actually asked:
What is the correct answer to “What is the probability that I will get cancer in the next ten years?”
This must be a Bayesian probability. You will not live the next ten years multiple times, so frequentism does not apply. There are ways of reframing this probability to be frequentist. For example, "What is the probability that a person who was [some age, health traits, etc.] ten years ago got cancer since that time?" - this is objectively answerable with actuarial calculations, and makes more sense in the context of frequentism than Bayesianism. If there were no objective answer to questions like this, then insurance companies would not be able to operate at a profit. However, the drawback is that you have to decide which traits should be included in the description.
Some traits are easy to measure (from the perspective of the insurer) and some traits are harder (think of age vs. diet). Some traits are more obviously relevant to cancer risk (think of smoking vs. skydiving). The insurer profits by focusing on traits that are either easy to measure, highly relevant, or both. So their probability is not perfectly applicable to you. But from the insurer's perspective, that's OK, because even though all of their probabilities are slightly inaccurate, in aggregate these errors should tend to cancel out on average, if the insurer has done a good job of picking the most important traits, and so they can still make a profit anyway. In this way, we can see that even this frequentist probability is subjective to a degree, because the insurer must necessarily accept a probability that isn't perfectly applicable to the problem domain. But, again, the insurer is not free to simply make up whatever probabilities it likes, or else it will start to lose money, because there is still an underlying fact of the matter. The insurer is simply accepting a less accurate rendition of that reality for business reasons.