One has to look at the cognitive process at work. What is its thought process, what is its motivation?
What is the function of this social manipulation (the cognitive process by which they arrived at social manipulation)? In that we'll understand what function(s) is/are at work.
If we go by what I said earlier of Fe and Fi, the former being about fairness, and the latter being about authenticity....
...then, using the crying example of my ESFP ex-roomie, if it was Fe-manipulation, it would mean that her aim was to bias fairness towards her favour. She had no acknowledged of fair, unfair; that wasn't her motivation. It was about validation of self. She was feeling like her self was infringed upon, being oppressed (by me), attacked, and she wanted some kind of validation that she mattered, her feelings mattered (she, as an individual, was valued). Because my attack was making her feel devalued. She was wanting someone to believe in her. My other roommates would validate such motivations spurred by her Fi.
She didn't go the route of expressing why one should believe in her, value her, no, she took that feeling spurred by Fi feeling infringed upon, and ran with it. And, what better way then to utilize Te, which, in her case, says (FiTe), hey, the fastest and most effective way to gain emotional leverage over others, so I can be validated, feel valued, is to bring on the waterworks, force it on the table. And, thus the crocodile tears ensued.
She had the emotional realm (F) covered, by Fi (more natural a function for her), thus she didn't need Fe to reach any conclusion. She wanted to, if I may borrow this phrase from Amar earlier in this thread, sync up her projected emotional state with my other roommates, which would validate her being. "Pain" for pain. I show "pain", they will feel pain (for me). And, Te helped it be projected externally. There was no need for Fe to come into play. Fi+Te would get the job done much faster, and more readily available to her.
In the end, the same conclusion is reached, social manipulation. Which Fe can reach too (all by its own). A lot of same conclusion (in terms of manifested behaviour, e.g., social manipulation) is reached by all types, but it doesn't mean, according to function theory, that it's all the same function, every time, that's manifesting said behaviour. One has to dig deeper - into the cognitive process - to unravel the functions at play.
Hence, why I called this example, Fi manipulation (not being authentic in her pain, yet wanting acknowledgement of its "authenticity"). Fi was the judgement making function at play. It was manifested (externalized efficiently) with the help of Te.