Is an acid catalyzed hydro-alkoxy addition the same as an acid-catalyzed alkoxyation?
Assuming by "acid-catalyzed hydro-alkoxy addition", you mean the alkoxy addition catalyzed by protonated water (
In an acid-catalyzed alkoxylation, the water is protonated by some strong acid such as
The alkene or alkyne breaks a pi bond, sends an electron for the proton, and deprotonates the water so that it's just water again, gaining one proton.
Then, the alkoxide, seeing the carbon adjacent to the carbon that just donated a pi electron as cationic and electrophilic, acts as a good enough nucleophile that it can attach onto the cationic, less-substituted site (standard, Markovnikov addition).
Then, the alkoxide, seeing that an unstable triangular pi-bond-banana-bond complex formed, attacks the less substituted carbon (standard, Markovnikov addition) and attaches to improve the stability of the molecule.
After attaching, the alkoxide is considered protonated (and cationic), so the previously-discussed, unprotonated water comes in and deprotonates the alkoxide, finishing the mechanism and regenerating the acid catalyst that began the reaction.