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".
In Modern English, according to the OED, access is defined, in this context, as "The power, opportunity, permission, or right to come near or into contact with someone or something; admittance; admission."
Entry, on the other hand, is defined as "The action of coming or going in," in this context (while the word does have a definition listed which is synonymous with access, that definition is considered obsolete).
In other words, to gain entry into the building means to get into the building. This can be done sneakily, or with permission, or even by brute force, barrelling your way in.
On the other hand, to gain access literally means that you have gained the power or "permission" to enter the building. You are being allowed into the building by the building's security (whether this security is a system of guards or a computer system), or you can reasonably get past the guard.
Obviously, this can be done in a number of ways, from hacking into a computer mainframe to access security keys, to pickpocketing a guard for literal keys, to dressing up and presenting yourself as a high-authority figure, to buying an admission ticket. However, at this point in time, if you have "gained access" to the building, you now have what can be essentially described as an unlocked and unblocked pathway into the building.
If you get in, you have gained entry.
If you are allowed in, you have found a way you can get in easily, or if you have gotten in and you pose little risk of being removed, you have gained access.
Best Answer
Satisfy has these two Latin words as a root (from Google):
whereas fulfill has this etymology:
When something is full, putting more in will make it overflow.
So satisfy implies that there is a minimum to consider good, but you could do more to improve things further than the minimum.
Fulfill implies that either you achieved a standard (reached the "full" line") or did not. There is really no such thing as "halfway fulfilled," for example.
This is why things like destinies or orders are fulfilled, but not satisfied. Either you reached your destiny or you didn't, either your order is complete or it is not. But you could satisfy an indignant customer who didn't get an order fulfilled by giving him/her a gift card, though a higher amount gift card would make him/her more satisfied.