To deal with math in an applied/science field you need to think in terms of reality and practicality and not only in terms of equations and proof.

what is the exact definition of the time-dependent firing rate

This is not really a question that makes much sense. There is no such thing as an exact definition here; the definition is whatever you choose to make it. But we can explain what Dayan and Abbott are talking about.

Let's think about what it means to have a time-dependent firing rate, physically. "Firing" is a discrete event, occurring at a moment of time with arbitrary precision. A firing rate is a probability, or more specifically a probability density, for that discrete event to occur.

There is no underlying reason for a probability density to itself be discrete, it can change at arbitrary time. That's why Dayan and Abbott write that the dT might as well be taken to the limit of zero. But we also expect probability densities for spikes to be relatively smooth, so there's not much lost if you use dT that is small but doesn't approach zero. So, practically, discrete is okay.

There's also not much need for spike times to actually be tracked with arbitrary precision. Spikes will result ultimately result in current flowing through channels into cells; cell membranes are capacitors with some time constant. If you're trying to model the effects of a spike, you're not going to be able to really tell the difference between time differences that are much smaller than those associated time constants. So again, practically, discrete is okay.

Finally, let's think about actual measurement and estimation. That's the motivation that Dayan and Abbott actually raise. If you want to estimate a firing rate from actual spike trains from actual data, you're going to have a finite number of events all at arbitrarily precise times. The first place you're going to lose the actual arbitraryness is going to be at the sampling rate of your recording equipment, so there's that. But even if that's not an issue, what happens to your estimated probability density as you get to smaller and smaller dT? You're going to end up with an estimate that is either 0, because you never ever observed a spike in that bin, or 1/nTrials. This isn't very helpful. I mean, it represents the data accurately, but it doesn't tell you anything. Much like you might summarize a sample of heights with a mean and standard deviation, the whole point of expressing some statistical quantity like firing rate is to be able to generalize and predict. So, it doesn't make sense to express firing rates based on data at a higher time resolution than we have actual data available to make a representation.

An alternative to Dayan and Abbott's approach of not letting dT be too small would be to convolve the discrete spike train with some smoothing kernel like a Gaussian. But to represent these things in a computer we're going to eventually end up with something discrete at some level.

For some cases, you might want to ensure a spike train is binary per trial for a choice of dT, but I wouldn't say it's necessary per se. A reasonable choice of dT will depend on the specific system you're modeling and what you're doing with it. For many computational tasks you'll also be dealing with computational constraints that may make a finer time resolution difficult to deal with.

To deal with math in an applied/science field you need to think in terms of reality and practicality and not only in terms of equations and proof.

what is the exact definition of the time-dependent firing rate

This is not really a question that makes much sense. There is no such thing as an exact definition here; the definition is whatever you choose to make it. But we can explain what Dayan and Abbott are talking about.

Let's think about what it means to have a time-dependent firing rate, physically. "Firing" is a discrete event, occurring at a moment of time with arbitrary precision. A firing rate is a probability, or more specifically a probability density, for that discrete event to occur.

There is no underlying reason for a probability density to itself be discrete, it can change at arbitrary time. That's why Dayan and Abbott write that the dT might as well be taken to the limit of zero. But we also expect probability densities for spikes to be relatively smooth, so there's not much lost if you use dT that is small but doesn't approach zero. So, practically, discrete is okay.

There's also not much need for spike times to actually be tracked with arbitrary precision. Spikes will result ultimately in current flowing through channels into cells; cell membranes are capacitors with some time constant. If you're trying to model the effects of a spike, you're not going to be able to really tell the difference between time differences that are much smaller than those associated time constants. So again, practically, discrete is okay.

Finally, let's think about actual measurement and estimation. That's the motivation that Dayan and Abbott actually raise. If you want to estimate a firing rate from actual spike trains from actual data, you're going to have a finite number of events all at arbitrarily precise times. The first place you're going to lose the actual arbitraryness is going to be at the sampling rate of your recording equipment, so there's that. But even if that's not an issue, what happens to your estimated probability density as you get to smaller and smaller dT? You're going to end up with an estimate that is either 0, because you never ever observed a spike in that bin, or 1/nTrials. This isn't very helpful. I mean, it represents the data accurately, but it doesn't tell you anything. Much like you might summarize a sample of heights with a mean and standard deviation, the whole point of expressing some statistical quantity like firing rate is to be able to generalize and predict. So, it doesn't make sense to express firing rates based on data at a higher time resolution than we have actual data available to make a representation.

An alternative to Dayan and Abbott's approach of not letting dT be too small would be to convolve the discrete spike train with some smoothing kernel like a Gaussian. But to represent these things in a computer we're going to eventually end up with something discrete at some level.

For some cases, you might want to ensure a spike train is binary per trial for a choice of dT, but I wouldn't say it's necessary per se. A reasonable choice of dT will depend on the specific system you're modeling and what you're doing with it. For many computational tasks you'll also be dealing with computational constraints that may make a finer time resolution expensive to deal with while not providing an appreciably different result.