Engage has a wide range of more or less connected meanings. You can engage an employee (give him employment), engage a gear (make it active), or be engaged to marry (pledged), etc.
The underlying senses of engage involve commit, pledge, become involved with, attach, deriving from the rare/archaic/obsolete...
gage: Something of value deposited to ensure the performance of some action, and liable to forfeiture in case of non-performance; a pawn, pledge, security.
Deploy derives from Latin displicāre (to unfold). Originally this was only normally used of military forces, troops, in the sense of to open out so as to form a more extended front or line.
By lately, particularly with reference to mechanically-activated weapons systems, the two terms have both been increasingly used in the sense of employ, use. It's a bit fanciful to say they actually mean anything different in OP's context, but if Captain Kirk on the Starship Enterprise were to issue the orders...
1: "Engage photon torpedoes!"
or
2: "Deploy photon torpedoes!"
...the first could be interpreted as an instruction to prepare the torpedoes for use, and the second as an instruction to actually fire them. The words are used so loosely that in the above context, they're really synonyms - but if you ask "In what sequence would Captain Kirk give these orders?" (a loaded question, implying there must be a difference), I suspect the vast majority of native speakers would say "1 then 2".
"Revocate" means to call back, recall. For example, to send a message to troops to retreat. I have an excellent vocabulary, but was previously unaware this word existed, so I think it's reasonable to assume its use will usually raise eyebrows among even very fluent, well-educated speakers — most of whom will assume you misspelled "revoke". Some dictionaries mark it as obsolete, and I'm not surprised.
"Revoke", on the other hand, is a fairly common word, meaning to undo a previously-valid decision or law. For example, to cancel a previously-granted permission to do something.
If you want to use revocate, you should seriously consider simply saying "call back" or something similar instead. Even in formal uses, this will likely be the best choice; borrowing from the previous example, a military text that mentioned a general "recalling" his troops would be perfectly natural, while one that mentioned that general "revocating" his troops would be hard to understand.
"Revocation", itself, has nothing to do with "revocate", other than by etymology. It's the noun form of "revoke". So one might refer to "the revocation of an order", meaning its cancellation.
Best Answer
When using the terms without other context:
To my [Southern US] ear, there is no strong semantic difference, although there is a slight difference in feeling. "Crossed" conveys slightly more tension, while folded sounds slightly more relaxed. Both sentences describe the same posture.
If there is more to the phrase, as in
then a different posture is described and the equivalence breaks down. I can cross my arms behind my back, but I don't think I can fold them back there.