Are nondeterministic algorithm and randomized algorithms algorithms on a deterministic Turing machine?
Broadly speaking, you can normally trade the word "algorithm" for the words "Turing Machine", in the sense that a Turing Machine is not a programmable computer like the one you have in front of you, it is a mathematical embodiment of an algorithm. So for the two cases you mention:


A nondeterministic algorithm is a nondeterministic Turing Machine.
A randomized algorithm is a probabilistic Turing Machine.