I'm a single 20-something man living in a major city who is actively dating. I met a woman about 5-6 months ago on a dating website, let's call her Alice. Alice and I hit things off well based on our mutual interest in early modern music and modern Japanese authors. We went on two dates to shows and had a great time each evening. However, soon we mutually decided we didn't want to pursue a relationship, but wanted to still be friends and continue to go to music shows and talks about books together. So we did; maybe seeing each other about once a month, and texting a handful of times a week to work to pick out shows, work out logistics, and occasionally share work/life updates in a way that's typical of friends.
Just recently Alice got a boyfriend, call him John. Last time I saw her, she expressed concern to me that she might no longer be able to be my friend because of the circumstances in which we met and what that would cause John to think. Nevertheless, she was willing to keep contact as a friend for now. Normally when I'm in this situation I'll try to arrange a double date or something similar so I can meet John and demonstrate I'm not a threat to him, and just generally bring things into the open. However Alice seemed less than thrilled at that idea. This made me a little uncomfortable that our friendship might be somehow secret, and poking at this further she explained it's just embarrassing to tell people, not exclusively, but especially her boyfriend, how we met despite the fact that we are just friends now.
What is an appropriate way to approach a conversation with Alice to ask her why she is bothered by telling people we met online? How can I ask her how we could make her feel more comfortable with introducing me to her social circle?
I would like advice on how to approach a conversation about how we can be friendly enough that we could introduce each other to our wider social circles without how we met getting in the way.