In usual Japanese slang (pretty much restricted to young men) Osu is a contraction for Ohayo gozaimasu, rather than arigato gozaimasu. OSU as used in Karate is, indeed Oshi-shinobu, (lit.: push and endure) and is used both as hai in the sense of " I heard and understand" and "let's go!". OSU should not be used to mean "yes": that would be "hai". Keep in mind that in Japanese, "hai" is not an exact equivalent to the english word "yes". OSU should never be used out of the dojo. If you learn your Japanese in a dojo, be careful about using the special language of the Dojo elsewhere. OSU would sound quite unpleasant to the hearer who is not a martial arts devotee.