>>94 No Skype.
A deposition is an advance form of cross-examination, but not conducted in front of the judge and jury. It is usually at one or other lawyer's office and it is recorded by video (what TS didn't want) and transcribed by a stenographer. The client will be there with his or her lawyer and so will the other side and his or her lawyer (hence her complaint about this guy being in the same room). The opposig side's lawyer does the cross examination. You are under oath when you make a deposition (that's what the article means by "in court").
Giving evidence by video link even in court in front of the judge is relatively rare. They want you there unless it is impossible (like you are a soldier on active service elsewhere). The point is that the the judge and jury needs to see your reactions. Similarly with depositions the point is that the other side wants to see how you will perform in cross examination. They would not tolerate her being over some video link (not least because her lawyers could be out of shot holding up a sign for her to read off).