What does "probability" really mean?

A probability is a quantified possibility. That doesn't tell you much about the ontology of probability, because it doesn't tell you much about the ontology of possibility.

A possibility could be a feature of reality, (a propensity) or it could be a merely apparent result of ignorance. (Knightian uncertainty). These are very different theoretically, but hard to distinguish practically.

Concrete examples of real possibilities include MWI branches or metaphysical possible worlds. Neither the meaning nor the usefulness of possibility depends on such realities. Ignorance is always with us, so possibility is always with us.

The quantification of possibility is a separate issue.

Probability is not always with us because it requires the extra element of quantification. Physics supplies no meaning or method for determining the probability of decoherent branches, for all that it is a topic of great philosophical interest.

https://www.lesswrong.com/posts/r282ErRKMFzxpKYMm/can-we-in-principle-know-the-measure-of-counterfactual?commentId=gMFvEMygeT6CTkqMa [LW(p) · GW(p)]

If you are interested in the objective probability of the coin flip,the it only has one value because it is only one event. In a deterministic universe the objective probability is 1, in a suitably indeterministic universe it is always 0.5.

The assignment of probability 1 to an event that has happened is also subjective. You don't know that it had to occur with complete inevitability, ie you don't know that it has a conditional probability of 1 relative to the preceding state of the universe. You are setting it to 1 because you are proceeding as if it has occurred. Although determinism is equivalent to the claim that everything happens with probability 1.0, the fact that 1's appear in probability calculations does not imply determinism.

If you think the questions “what will it be” and “what was it” are different, you are dealing with subjective probability, because the difference the passage of time makes is a difference in the information available to you, the subject.

Failing to distinguish objective and subjective probability leads to confusion. For instance, the sleeping beauty paradox is only a paradox if you expect all observers to calculate the same probability despite the different information available to them. The trick is to drop the assumption that there is a "the" probability everyone has to agree on.

The ways possibilities are quantified varies, too. Bayesianism and frequentism are different methods of quantification that often agree. Bayesianism permits a subjective element without excluding objectivity. Frequentist assumptions can be used as priors, and Bayes can converge in frequentist conclusions.

The objective basis of frequentism is the measure of an equivalence class. The objective basis of frequentism is not an objectively existing possibility, ie. frequentism is defineable in a deterministic universe. However , for a measure to be translated into a probability, something has to happen. An urn might contain a certain ratio of black balls to white balls , but that is not a (frequentist) probability, until you imagine a ball being drawn at random. Randomness, even imaginary randomness, introduces the necessary element of multiple, non trivial possibilities.

Objective and subjective probability are not mutually exclusive opposites. Strict determinism does not imply any level of information for any observer, so observers can still apply probabilities based on their ignorance of how things turn out

All concepts, including probabilities, are tools for making better predictions of the anticipated experiences. One can discuss whether a single universe, a Tegmark multiverse, MWI or something else is a more convenient and consistent way of describing "reality", but that has no bearing on "what probability really means" or "what electromagnetic field really means" or "what consciousness really means". All these are concepts (models) that are useful within their domain of applicability, and not very much outside of it. There is no one universal concept that is always useful in every case.

If you want some philosophical foundation, I suppose the first step is to decide whether probability is in the mind [LW · GW], or in the territory.

The MWI/Tegmark approaches are trying to find probability in the territory; to define it as a frequency among some possible worlds.

The opposite approach is to define probabilities as a tool to deal with uncertainty. You can have uncertainty regardless of the underlying structure of the universe. (You could even have uncertainty about the underlying structure of the universe.) If thinking creatures can evolve in Conway's deterministic Game of Life, they too could be uncertain about some stuff, and could invent probability to describe their uncertainty.

There is also the somewhat boring answer that probability can refer to anything which obeys the axioms of probability.

Does this help?

Be careful with ideas of what "most LWers" or "the LessWrong community" believes.  There's a LOT written here, some of which is useful, some interesting, and some kind of pointless.  The fact that something gets explored and not specifically refuted may only mean that it's fun to explore, and the scepics don't find it worth the effort to engage.

I'm pretty agnostic on MWI.  Not because I have strong counterarguments, but because it's hard to know what "true" or "real" even means outside my light cone.  Fortunately, it doesn't matter.  Bayesean probability is about prediction or knowledge of future experiences, not about reality (whatever that is).

I subscribe to the Jaynes/Laplace view of probabilities, namely that they exist in the mind and result from changes in information rather than changes in the world, let alone multi-worlds.

Imagine I tell you about an urn with black and red balls, without an additional detail. You can provide a probability of getting a black or red ball (50/50). 
As I provide more information ("there's 5 red balls but 50 black balls", "the red balls are sitting on top of the pile", ...) your probability assignment will change without the physical urn having changed at all. 
As your knowledge of the urn and the selection mechanism becomes more complete, your uncertainty decreases and your confidence/probability levels grow.