The Angel ep is called "Dear Boy."  Here's a snippet: 
Darla and Angelus exit the alley and walk along the street.
Angelus: "What is it?"
Darla:  "Surprise. You have to guess."
Angelus scans the street. Finally spots a family with three young daughters coming out of a building.
Angelus: "The three daughters - all virgins."
Darla:  "Close."
Angelus:  "The one in the middle has something delicate and unique… Did you find me a Saint?"
Darla:  "Better than that - she has the sight."
The girl in the middle turns. It's Drusilla. She looks straight into Angelus' eyes.
Angelus:  "Visions. She sees the future. (Angelus, Darla clinging to his arm, starts walking towards Dru) She is pure innocence, yet she sees what's coming, she knows what I'm going to do to her.  I'll really have to come up to snuff for this one."
Dru turns away from them and herds her sisters away.  Darla puts a hand on Angelus chest to stop him from following further.
Darla:  "Down boy, let the plum ripen."
Angelus:  "You always come up something new."
Darla:  "Keeps me young."

Now, there's another scene when Anelus drives her insane after killing her family, and in the transcripts, it keeps referencing her as Dru, so I assume, that has always been her name.  Or at least, some variation of it.  I think it would make more sense that they had her keep her real name to add to her torment.  That was Angelus' intent anyway when he created her- eternal torment (so he says later in the episode).  And Spike doesn't seem to have officially changed from his official name till he killed his first Slayer.  Or at least, his vampire family don't acknowledge him as such until that point (so says canon- particularly on Ats s5).

Hope that helps :D

Oh, and since it means "fruitful or dewey eyed" in English (so says wikipedia), I'd say Darla, only because Angel's Irish.  Unless they just meant English period, and not British English...then it could've been Angelus who gave her the name, if it wasn't her original name (which they never officially say in the episode).

When it came to Anyanka? Angel Darla and Spike ex. they always made a great point of either pointing out their real names, or how what they are being called isnt their real name (Darlas case) there is never any confusion when it comes to Drusilla which makes me believe that is her real name esle there would be some indication. (typical buffyverse modus operandi)

I would believe Drusilla to be her human name. She might have been christened something else, as was common with Catholic and Anglican families, with "Drusilla" being her middle name, but it does seem that Drusilla is what she's always been called. Spike and Angel both call her Dru later, at times, but I don't think Darla did. I can't remember her ever shortening the name.

  need to rewatch AtS ... going to update my Netflix account and bump all of my husband's Robot Chicken eps down in the queue ...

Good plan, Verity.  Of course you could just sign up for that new full set of Angel when he isn't looking.

Thing is, from the snipplet you posted, you're talking of non-dialogue references. Since the show never mentioned a different name for Dru, of course the transcripts refer to her by that name, otherwise no one would know who they are talking about. Especially if the transcript in question was written by some fan; the non-dialogue parts are simply descriptions of what they saw, so if they see the character they only know as Drusilla, that's what they'll call her. But even if it was an original transcript - be Drusilla her real name or not, if the writers didn't bother to come up with a different name, they can hardly refer to her by that non-existant name.

Take Darla for example. We know Darla isn't her real name, but I'm sure if you were to dig up the transcript of the eppisode/scene where we meet her human self, it will refer to her as Darla even before the Master shows up - simply because that's the only name we know for that character.

Btw, since nothing was ever said to imply that Drusilla was "renamed" after being vamped, I'm inclined to assume it's her real name. Don't think it really matters too much anyway, unless you want to write a fic about pre-vamp Dru and her not very religious/catholic sounding name poses a problem. For anything else, I'd stick with what the readers know from the show - after all, be it her real name or not, if the show didn't feel the need to answer that question in seven seasons of BtVS plus five of AtS, why should you have to.

I don't know where or when but I did hear that Dru's human name was Mary. It's possible that was a fanfic writer's idea and I just liked it so much that I remembered it. But still, it would make sense for me - Mary the Saint Virgin, turned into insane Drusilla, the sister and lover of her even more insane emperor brother...

Oh, I see I'm not only one Mary's hunting, huh? I'm sure this flashback will be great. Good luck!

I have read, fics where they say Drus middle name was Drusilla and her first name was Edith, that is why she has miss edi.

I must have missed the talk of Darla's real name because I always understood it was Darla.