As already noted by others, there is no specific Bayesian definition of probability. There is only one way of defining probability, i.e. it's a real number assigned to some event by a probability measure, that follows the axioms of probability. If there were different definitions of probability, we wouldn't be able use it consistently, since different people would understand different things behind it.
While there is only one way we define it, there are multiple ways to interpret the probability. Probability is a mathematical concept, not related anyhow to real world (quoting de Finetti, "probability does not exist"). To apply it to real world we need to translate, or interpret, the mathematics into real world happenings. There are multiple different ways to interpret the probability, even different interpretations among Bayesians (check Interpretations of Probability in Stanford Encyclopedia of Philosophy for a review). The one that is most commonly associated with Bayesian statistics is subjectivist view, also known as personalistic probability.
In subjectivist view, probability is a degree of belief, or degree of confirmation. It measures how much someone considers something believable. It can be analyzed, or observed, most clearly in terms of betting behavior (de Finetti, 1937; see also Savage, 1976; Kemeny, 1955):
Let us suppose that an individual is obliged to evaluate the rate $p$
at which he would be ready to exchange the possession of an arbitrary
sum $S$ (positive or negative) dependent on the occurrence of a given
event $E$, for the possession of the sum $pS$; we will say by
definition that this number $p$ is the measure of the degree of
probability attributed by the individual considered to the event $E$,
or, more simply, that $p$ is the probability of $E$ (according to the
individual considered; this specification can be implicit if there is
no ambiguity).
Betting is one of the situations where one needs to quantify how "likely" he believes something to be and the measure of such belief is clearly a probability. Translating such belief to numbers, least to measure of belief, i.e. probability.
Bruno de Finetti, one of the major figures among subjectivists, notices that the subjectivist view is coherent with axioms of probability and it needs to follow them:
If we acknowledge only, first that one uncertain event can only appear
to us (a) equally probable, (b) more probable, or (c) less probable
then another; second that an uncertain event always seems to us more
probable then an impossible event and less probable then a necessary
event; and finally, third that when we judge an event $E'$ more
probable then event $E$, which is itself more probable then an event
$E''$, then event $E'$ can only appear more probable then $E''$
(transitive property), it will suffice to add to there three evidently
trivial axioms a fourth, itself of purely qualitative nature, in order
to construct rigorously the whole theory of probability. The fourth axiom
tells us that
inequalities are preserved in logical sums: if $E$ is incompatible
with $E_1$ and with $E_2$, then $E_1 \lor E$ will be more or less
probable then $E_2 \lor E$, or they will be equally probable,
according to wherever $E_1$ is more or less probable then $E_2$, or
they are equally probable. More generally, it may be deduced from this
that two inequalities, such as
$$ E_1 \text{ is more probable then } E_2,\\ E_1' \text{ is more
probable then } E_2',$$
can be added to give
$$ E_1 \lor E_1' \text{ is more probable then } E_2 \lor E_2' $$
provided that the events added are incompatible with each other ($E_1$
with $E_1'$, $E_2$ with $E_2'$).
Similar points are made by multiple different authors, like Kemeny (1955), or Savage (1972), who like de Finetti draw connections between the axioms and subjectivist view of probability. They also show that such measure of belief needs to be consistent with the axioms of probability (so if it looks like a probability and quacks like a probability...). Moreover, Cox (1946) shows that probability can be thought as an extension of formal logic that goes beyond binary true and false, allowing for uncertainties.
As you can see, this has nothing to do with frequencies. Of course, if you observe that nicotine smokers die of cancer more often then non-smokers, rationally you would assume such death to be more believable for a smoker, so frequency interpretation does not contradict the subjectivist view. What makes such interpretation appealing is that it can be applied also to cases that have nothing to do with frequencies (e.g. the probability that Donald Trump wins the 2016 US presidential election, the probability that there are other intelligent lifeforms somewhere in the space besides us etc). When adopting subjectivist view you can consider such cases in probabilistic manner and build statistical models of such scenarios (see example of election forecasting by FiveThirtyEight, that is consistent with thinking about probability as measuring degree of belief based on the available evidence). This makes such interpretation very broad (some say, overly broad), so we can flexibly adapt probabilistic thinking to different problems. Yes, it is subjective, but de Finetti (1931) notices that as frequentist definition is based on multiple unrealistic assumptions, it does not make it more "rational" interpretation.
de Finetti, B. (1937/1980). La Prévision: Ses Lois Logiques, Ses Sources Subjectives. [Foresight. Its Logical Laws, Its Subjective Sources.] Annales de l'Institut Henri Poincaré, 7, 1-68.
Kemeny, J. (1955). Fair Bets and Inductive Probabilities. Journal of Symbolic Logic, 20, 263-273.
Savage, L.J. (1972). The foundations of statistics. Dover.
Cox, R.T. (1946). Probability, frequency and reasonable expectation. American journal of physics, 14(1), 1-13.
de Finetti, B. (1931/1989). 'Probabilism: A critical essay on the theory of probability and on the value of science'. Erkenntnis, 31, 169-223.
