The balance, handed over on the doorstep
The remaining half of the fee is handed over when she reaches your room, in cash or by the routes already agreed, and either of you can pay it.
The balance is whatever is left after the deposit: the second half of the fee, settled at the door when she arrives. There is no running total during the visit, nothing is added for the journey, and no figure is negotiated on a doorstep.
What it looks like in practice
On ninety minutes for the two of you with one therapist, the half settled beforehand and the half handed over are the same size, so the door takes exactly what the deposit took. On two hours with a therapist each the balance is a larger amount of money but still one half of the whole, and it changes hands once she is inside. The confirmation in the thread already named both halves, which is the point of having it in writing.
Currency
Every figure quoted is a euro figure. Cash at the door in another major currency is welcome, and when that is what you intend, the exact sum in that currency is confirmed to you in writing before she sets out. Nobody should be converting anything on a phone at eleven at night.
Which of you pays
Either, or both, and the house has no view on it. The deposit and the balance are separate settlements and nothing connects them, so a couple splitting a trip between two cards can split this the same way. What is not possible is paying the whole fee in cash in advance, since the deposit is settled before she leaves.