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In general, the on/off delay is how long it takes from changing the input until the output starts to do something.  The rise/fall time is how long it takes to do the something, once it starts doing it.

Your first FET driver datasheet is a good example:

<img src="https://electrical.codidact.com/uploads/u9gPVMXNiFRKqjcCcQ7qY5cB">

The switching characteristics are idealized as up to 300 ns where nothing happens after you change the input, then another 30 ns for the output to transition on/off or off/on once it starts doing something.  The total worst case time from switching the input until the output is fully in its new state is therefore 330 ns.

So why would anyone care about the details and not just the overall 330 ns delay?  Because sometimes the overall delay isn't as important as a fast transition.  An ideal switch dissipates no power when off because the current is zero, and no power when on because the voltage across it is zero.  The power dissipated in-between can be significant.  You want to have the FET transition thru the in-between region as fast as possible.

This FET driver guarantees that it won't leave the FET gate in the in-between region for more than 30 ns each transition.  If you're switching a high power heater, for example, the 330 ns delay from when the thermostat says to switch until the load is actually switched is inconsequential.  However, transitioning the FET quickly can be the difference between requiring a heat sink or not.

The specs for a FET are interpreted similarly:

<img src="https://electrical.codidact.com/uploads/JBKStPJhWi78yY3Uf6VETnY5">

The delay time in this case is how long it takes for the channel to change its conduction characteristics from a gate voltage change.

The rise and fall times is how long it takes for the drain voltage to change in a particular circuit.  Note how it explicitly says the off/on gate voltages are 0 and 18 V, and that the load on the drain is 2 &Omega;.  These are probably the conditions under which the manufacturer knows the response will be fastest.