BUT soon afterwards the second time, that would be about 3 weeks ago, my friend called me on skype, explaining that her partner didn't want me and her having emotional relationship. I also talked with him, to get what he meant too. He said that he was very fine with me and her being best friends that have sex from time to time but he didn't want us being lovers. (And he doesn't have any objections for continuing the 3somes.)
Could be up front with him.
"Thank you for your input. Could you be willing to explain what about us sharing loving feelings as well as sex that is hard for you?"
If it is is something that could be articulated and resolved -- great. Resolve it. If this is his only objection...
When he was asked about it, the response was that he didn't like us having public displays of affection such as holding hands or cuddling while watching movies
Then whether you understand that or not -- you could ask if that is a soft limit (could change in time) or a hard limit. (will never change) Decide if that's a price of admission you are willing to pay.
If you wonder this...
I thought it might be that he when he saw us together it made him see that there's definitely a deep, meaningful connection between I and her and that got him scared somehow
Could ask up front if that is what he's experiencing. He was scared to see that deep connection between you and it riled up jealousy from fear of abandonment.
If it is something else he cannot explain, then the ethical thing seems to be to break it off with her and be honest with both. Maybe something like...
"I'm sorry. I cannot agree to those limits. I already have loving feelings toward her. I think we best take a time out, stop everything here. And decide how we want to be. I would like to be with her as her friend and lover. Show of hands -- who is willing/able for that to exist?"
If all the hands don't go up, this polyship "V" with her as the hinge isn't a runner unless something changes.
- BF becomes willing and able.
- GF breaks up with BF.
There really isn't anything YOU can do here that I can see other than step away for a while to let them figure out his side and her side.
I do not suggest you keep sharing sex with her and threesomes with him because "it's better than being alone with nothing" because that would not be self respecting behavior. Over time that starts to pinch and becomes a breeding ground for resentments.
Tread carefully here.