I think you will frequently find that measures in biology are not defined in a consistent manner, largely because experimental constraints often require a different operational definition. People prefer to keep the same term for the underlying concept, but recognize that different operational definitions may give slightly different answers. I'd consider this in the realm of "all models are wrong, but some are useful".
In the case you describe here, I would argue this is a conflict between a theoretical and practical version of the same concept.
For drugs, the concept of bioavailability generally applies to different dosing strategies and contrasts intravenous injection (bioavailability = 100% because the entire injection is immediately introduced to systemic circulation) with other routes (e.g., oral, transdermal).
Operationally, though, people don't typically measure bioavailability by counting individual molecules in the systemic circulation versus not. Instead, they measure serum concentrations at multiple time points, calculate the area under the curve, and use a ratio of area under the curve to determine bioavailability.
Here's an example of how that might look from Wikipedia:
For the IV curve, you just have decay kinetics; for PO you have absorption and decay. To compare them, you integrate to get area under the curve which has units of concentration*time, and look at the ratio of PO to IV to report PO bioavailability.
Sort of the "null hypothesis" or basic assumption for kinetics in pharmacology is that everything is single exponential unless you find otherwise. That is, rates of absorption and clearance are directly proportional to concentration in different compartments, or in other words every molecule behaves independently like you expect for any process driven by diffusion or unsaturated enzyme kinetics. Under that assumption, this AUC ratio is equivalent to "fraction of an administered drug that reaches the systemic circulation" and it's also true that "Bioavailability does not take into account the rate at which the drug is absorbed": it doesn't matter "when" a molecule enters the systemic circulation, if you calculate AUC you will at some point have that molecule enter and then leave and it will spend exactly the same amount of average time in circulation as a molecule entering at any other time.
However, drugs aren't always going to perfectly obey exponential kinetics, and another meaning of bioavailability is just "area under the curve", whatever the contributions to the area under the curve might be. That might include a drug that enters circulation very fast saturating an enzyme that metabolizes it, leading to a slower clearance for a drug absorbed quickly versus slowly. If that's the case, your AUC ratio no longer precisely represents bioavailability in terms of "fraction of an administered drug that reaches the systemic circulation" but it retains a meaning of bioavailability in terms of "dose*time".
Your definition:
Bioavailability (F) is defined as the rate and extent to which the active constituent or active moiety of a drug is absorbed from a drug product and reaches the circulation."
is consistent with this practical measurement issue from AUCs.
In contrast, if you are using some sort of pharmacological model with explicit terms for absorption, decay (including higher-order decay terms), etc, you would want to use the theoretical definition for bioavailability.
I'll add that when you read:
Bioavailability does not take into account the rate at which the drug is absorbed
I would interpret this as trying to emphasize that bioavailability is not about peak concentration. In medical practice, this is very important to understand, whereas in a pharmacokinetic sense the other nuances may also become important.
