Well it looks like you got useful feedback, but I'll add, yes I'd like to know if stuff going on in my partners relationship is happening if it is about me or affects me and I think in a triad it certainly would affect me. Otherwise arguments or bickering between who I date and their partners aren't my business unless they choose to confide in me.
And I sure think you should tell her, because its obviously on your mind, as well as G's desire to alter your agreement about privacy as it suits. It is "polite" to tell G ahead of time, because you explicitly agreed not to, but so I'd feel obliged to say I was going to and follow through. I would do it but I'm sure I'd be nervous they'd try to change their mind or get arguments ready to defend themselves, not sure just what order I'd do things in if I was in a triad, but I would certainly let K know.
K is your friend as well as your lover, and you'd tell a friend if their partner had been cheating I imagine? If you have forgiven, that should come through, so you aren't telling them in order to screw things up, just to inform. Also, if K is a supportive partner as you seem to say, she can be there to support both of you, and might find it not fair that she wasn't allowed to be informed and there for you.
On a side note I went and read those threads. I hope G is acting better now and you'd negotiated some boundaries she can stick to if she's agreed to them.
Happiness will never come to those who fail to appreciate what they already have.