Now, exact mathematical definitions for such speeds are difficult to have (maybe with hyperreal numbers... who knows). They go beyond the typical mathematical models for such things and with that beyond physics-based definitions of speed.
Still, I think Inaccessible speed and infinite speed in the suggested definition can reasonably be demonstrated to be the same thing.
To see why, let's look at an example: Say someone can run an infinite distance in 10 seconds. We assume the runner does so at a constant speed, meaning that whenever he runs 1/n-th of the time he ran, he covered 1/n-th of the total distance he ran. (Or, equivalently, that if he runs x-times longer he covers x-times more distance)
With that in mind let us ask ourselves the following question: How long did the runner need to cover the first 100m of his run?
Let's assume that there would be some timeframe t seconds that is longer than 0 seconds in which the runner runs the first 100m (let me add for the math guys here that t shall be a real number).
The runners full running time from 10 seconds, is (10/t) times longer. 10/t can be an incredibly large number, but since t is greater 0 it can't be infinite. (specifically: the division of 2 real numbers unequal 0 is a real number)
Now, by our assumption of constant speed we know that when he runs, for example, twice as long he would cover twice as much distance. So in 2*t seconds he would have covered 2*100m = 200m of distance running.
Similarly, in 10 seconds, which is (10/t)*t seconds, he would have run (10/t)*100m. That could once again be a very large distance. However, as both (10/t) and 100 are finite numbers it can't possibly be infinite distance.
Therefore we have found a contradiction, as we postulated that the runner ran an infinite distance in 10 seconds.
As we gain a contradiction like this the one assumption we made, namely that there is a timeframe t seconds that is longer than 0 seconds in which the runner runs the first 100m has to be false.
As such, the timeframe in which the runner ran the first 100 meter can only be 0 seconds. Anything greater than that leads to a contradiction.
What this demonstrates, in general, is that any character capable of running an infinite distance in finite time is also capable of running a finite distance in 0 time. So infinite speed implies inaccessible speed by the proposed definitions of the terms. As, the other way around, inaccessible speed implies infinite speed the term are equivalent, meaning that they are in fact one and the same speed ranking.