'Uruks' and 'Uruk-hai' are not two very different things (at least as the terms are used in the later part of the Third Age).
The word 'uruk' at that time refers to the great soldier-orcs of Mordor and Isengard. 'Uruks' is merely an anglicized plural. 'Uruk-hai' would seem to be the group plural most likely meaning something to the effect of 'Orc-folk'.
And yes both Sauron and Saruman had Uruk-hai.
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Tar-Elenion
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