Common mistake

I recently came in thought of something, something almost everyone here does.
I'm not sure if people are aware of it being a mistake or if they never thought about it, in my case I had just really thought about it because I was replying with someone and when I started they "poked me"
So I thought why not share it with everyone.

When we plot to rp we always ask our partners whatever they wish to rp in 1st or 3rd point of view, or we mention specifically what we prefer.
The thing is I guess what people really mean is if they want to rp script-style or writing-style. 
I have seen many people say "I want to rp 1st please" and then we start and they send a reply like this.

"*he moves toward her and holds her hand then says* your hand is so soft"

This is actually not a 1st p.o.v, this is a script-style 3rd p.o.v

How I came to think about it was like this. I was rping with someone, we come with a very interesting plot and I asked what they preferred they said 1st and I wrote my starter. I'm not going to write the same just a small example below.

My starter:
I wasn't sure what to do, the situation was getting bigger than me, I never intended for him to find out this way, I wanted to be alone with him, and slowly and reasonably start explaining it to him. I wanted os to  solve it like two adults, but with the furious look he was sending across the restaurant I knew deep inside me it wasn't going to be anything  adult-like of civilized(...)

So after sending my starter, my poor partner felt confused and said: "I thought we agreed on 1st pov not third", and his comment confused me too, because when I read my starter for like five times, I could only think of it being 1st p.o.v. not 3rd. But then I got it when they sent their reply. They meant script-style...

So now, I'm wondering if it's only me who had been in such a situation if it is actually commonly known or common mistake here to name the writing style with wrong phases.