One: the power of a name is confirmed not to be a thing in this setting. Telling her our name doesn't really give her much. Especially because I am pretty sure that we have probably abandoned our last name seeing how neither Una or Cerys has it on their character sheet. I also don't think either of them actually want to be associated with their parents anyway, which is probably why it would be abandoned.
Also since Morgan only gave her first name, we'd probably only reply with our first as well.
While Cerys is not a common name being able to link Cerys currently in the US to the Cerys born in ashford england is somewhat slim.
Two she just mentioned that there is some things she can't answer pertaining to oaths and trying to fish information out of her on the oath (even if it is just confirmation or denial of a particular clause) is slightly shady. This however isn't really the biggest issue at all.
Third what you say effects the conversation the more conservative and paranoid we seem, the more the conversation will be coloured by that. If we are trying to make friends with her, colouring the situation with more paranoia, or trying to dig into possibly personal things won't help at all.
Fourthly: Morgan is both "The one who wanders" and a person with a good grasp on the magical world. She has at least travelled enough to see multiple very powerful and very old magical beasts, and is inquisitive enough to check magical signatures and explore the area. It isn't hard to imagine that she actually contacts the magical community often.
This is a chance: A chance to make an impression on the magical community using Morgan as a vector. Right now we are "Strong, can communicate with Spirits of the land, seems untroubled by there being a strange creature in the water, killed a demon."
I don't particularly want her to keep quiet. This is a chance to start making a reputation in the magical community, and having a friendly relation with her might actually mean that we can more easily contact, and be respected by the magical community in the future.
Also trying to nix her talking about this situation, and about us when we were the ones that revealed a demon has touched down in that town is a pretty big negative. Anything that increases the chances to reveal to the world (or at the very least to the greater magical community at large) that the demons have stumbled on to Earth is big.
Also I kind of hope, that if we decide to burn down the forest, we tell her that before she leaves.
More importantly that we tell her that the spirit of the land told us to do it and guarrenteed that no humans would be harmed so that the act at the very least isn't attributed to active maliciousness to the people she tells about it, and since she probably contacts the magical community that is very necessary to at least try to avoid the risk of having not just the mundane but the magical communities breathing down your back.
And having someone who can say "oh yeah she told me about that and her reasoning to do so before she did it" is pretty useful if you need to convince people that the reasoning you gave wasn't just an excuse.
Hence, I am not going with the paranoia route because it can colour the conversation negatively, I want a good relation with her, giving first names isn't really overtly harmful in this settings, and because it implies denying her the ability to tell people about this when I actually want some people to know because it would be useful in the long run for some things to be known.