RL: Randomized Logarithmic-Space Has the same relation to L as RP does to P. The randomized machine must halt with probability 1 on any input. It must also run in polynomial time (since otherwise we would just get NL).
While requiring runtime in polynomial time avoids trivial RL=NL, it also allows running time stored. If an algorithm would on average take $f(n)$ steps, then trying to run $100f(n)$ steps would only have 1% chance of not halting, and rejecting in this case doesn't harm.
So why only "Contains RHL" shown?
What does "allows running time stored" mean?
Logarithmic-Space allows storing polynomial-size integer
What do you mean by "rejecting doesn't harm"?
If X is in RL, then there's an algorithm that accept for 51% chance for input in X, and we can construct one that accept for 50% chance and never fall into infinite loop
Why do you think the two classes should be equal? Their definitions are different.
RHL: Randomized Halting Logarithmic-Space
Has the same relation to L as RP does to P. The randomized machine must halt for every input and every setting of the random tape.