I do think 'keeping secrets' can be complicated, and quite agree that you cannot ask someone to 'keep their own secret'. I think that OP is upset because she feels they will be ridiculed by her cousin, and her DH just doesn't understand that.
I always thought that I could 'keep a secret' and indeed have kept many. But a close relative told me something in confidence that not only shook me dreadfully, but affected me personally. I felt I needed to deal with that aspect, and I told 2 people: DH & a trusted friend. They have not betrayed it, but I was so much better for talking about how it affected me.
I have however, never told the relative this. She regarded it as 'her' secret and I think had no idea of how it upset me. I think that the only other way I could have dealt with it was by going to a counsellor (which I considered)
And although some replies, in gransnet manner, were sharp, they were not 'trolling' at all!