You didn’t want a door on the shower and you have exactly that. If you want to add a door, you pay.

I agree with Joseph. That's part of what you are paying her for - her experience. The only caveat is if she had a discussion with you where she warned you that the design would spray water all over the place without a door, and you chose to proceed anyway. If so, then the responsibility is on you. Otherwise, she needs to learn this lesson the hard way.

I agree with you that shower doors make one feel somewhat closed in, especially for a taller or larger person, and they are a hassle to keep crystal clear.

It is wonderful to shower in a wall-free space.

The wall-free showers I have been in seem much larger than yours, although it is difficult to discern the measurements of your space in a photo.

I am sorry you got the proverbial pig in a poke with this interior designer.

She may have a degree but not that much experience, especially with bathrooms.

There is a sub-specialty of interior designers that are certified kitchen bathroom interior designers that have extra training and experience in kitchens and bathrooms and pass examinations, etc.

Just in case you get over the disappointment of this re-model and think about doing another bathroom or your kitchen.

I hope you come to an acceptable solution.

That's really a shame that it isn't what you wanted. I wouldn't want a door simply because I don't want to keep the glass clean, but since your design already has glass you need to keep clean, I might just add a door and be done with it. I'd ask for some of my money back from her for sure.
Your tile is beautiful!
I can't really see the shower hardware because of the glare/reflection of the light, but maybe it would be possible to change out the hand held for a rainshowerhead on an arm (or add a unit with both). Perhaps with the water flowing down instead of out (as I'm assuming it is) and a gentler stream of a rainshower, it might minimize splash.

No one has asked the OP if the designer mentioned the possibility of needing a door. So often, on this site, we see people ignoring solid, practical advice. Once someone has their heart and mind set, it is futile to attempt to persuade them to change their mind. I often experienced this at work. Usually, people get upset or even angry if you raise common sense questions about their design decisions. “I know. But that’s what I want.” People tend to hear what they want to hear and forget the objections raised by someone else.

I do believe the designer has some fault, he/she should have let you know that water would get out. The glass panel is not wide enough for your design thus he/she should have put a door on it. We have done shower like this that only have a panel but the glass panel is 48” wide and even then we do have minor splash outside the shower.

Presumably you okayed the plans before they were built. Seriously, Blind Freddy could see that the water was never going to stay within the confines of the shower given the size it is. You would have had more of a chance if the shower space included where the toilet is, and the shower wall extended from the right hand wall to at least where the toilet starts, with the entrance to the shower where the toilet is now.