Frequency and rate are interchangeable and are defined as observations or counts per unit of time. A count is the actual observation. Probability models are the mathematical representation of the frequency or rate. For example: probability of event A = P(A) = number of counts of outcome A divided by the set of all possible outcomes (S). I hope this helps :)

Count and frequency are synonyms. If the count is expressed per unit time or space it is called a rate. A probability model which gives the probability of the event of an accident is called a probability distribution.

Accident counts can follow a binomial probability distribution if the sample size is fixed. If the count is in an interval of time or space and the mean rate of occurrence is known the count can follow a Poisson probability distribution.

Asking complex question is a good custom to set the problem precisely, which is not always possible or at least very difficult in terms of everyday language. In the current question the bad guy is the word model. Adding explaining adjective-like words frequency, probability, rate - the first impression is that the problem is to compare the implied three (in the question - even four) types of models. As usually, the answer is dependent on the "school" using these notions.

Some examples of usage of the words:

a frequency of happening of a particular feature A among some population is the ratio n(A)/n where n is the size of the sample taken to this measurement and n(A) is the number of elements with feature A among the sample elements;

a frequency of the event A during time interval of lenth T is the ratio n(A)/T;

a rate of increase of quantity x with respect to y is the derivative dx/dy;

probability of a result A is sometimes just a theoretical quantity assigned to A jointly with many other results called events in the probability theory; simultaneously some rigorouslu determined conditions are requaired like positivity, additivity and normalization.

As it is seen from the above there is no simple answer to the question but some description of cases named by specialists not always in the same way. Obviously, one can imagine a subdomain in science where the notions in question are strictly defined (say, in the mass service modeling, in reliability theory modeling, in statistics of evolution of populations, in chemical models of some complex reactions between many types of agents, in physics where discrete and continuous words coexist, . .

Sumarising, probably there is no strict range of meaning of the notions asked within the current question without more detailed specification of the phenomena to be modeled.

Best regards, Joachim