I think there are two main things that stand out for me.
First, in a babysitting situation, you have to trust the person you are asking to be responsible for your children. With anyone (whether you pay them or not), you have to trust that they will follow your explicit directions/rules, that they will behave in a reasonable way when confronted with a situation that was not explicitly addressed in advance, and that they can handle unexpected situations appropriately. If you don't have that trust in a person, you absolutely should not be leaving your children in their care.
With a friend who is babysitting as a favor, I also think that you have to limit your explicit directions/rules to only those things which would be highly problematic for you if done contrary to your wishes. With someone you're paying, you can make any demand you like (and since your relationship is a purely business one, their perceiving you as being demanding has no social consequences). With a friend, especially one you're not paying, being overly demanding will almost certainly affect the friendship.
And, I almost think it goes without saying, but you have to actually state explicitly any directions/rules that really matter to you, and you can't expect someone else to just intuit them (no matter how self-evident you think a given rule is).
Second, the errand Michelle left the house to run was related to her livelihood, and it sounds like it wasn't something she necessarily knew about in advance. Especially if she didn't know about it in advance, I don't really see how she could have done things differently. It's entirely possible she didn't have time to notify Lynn and potentially wait for Lynn to come collect her children. And unless she had time to wait for Lynn to come and pick up her kids, telling her about the errand as she was leaving the house doesn't really serve any real purpose.
I also am of the opinion that if you're going to entrust your children to a person, you have to either accept that any adult in their household might end up with responsibility for them, or you have to explicitly say that you are not okay with specific adults doing that. And honestly, if you can't or won't trust all of the adults to be solely responsible for even a brief window, you shouldn't be putting your kids into that environment. That kind of goes back to the first point.
Regardless, Lynn can feel however she likes about the situation as it went down, but if she made it clear to Michelle that she was angry, then she overreacted. I don't think Lynn was wrong, because I'm not sure she could have anticipated exactly what happened, but for the same reason, I don't think Michelle was wrong, either. Clearly, Michelle is not a good caretaker for Lynn's children, so the best response would have been for Lynn to quietly resolve to never leave her kids with Michelle again, and for Michelle to quietly resolve not to watch Lynn's kids again. Even if Michelle could guarantee that she personally would be present at all times in the future, I think the situation showed that what Lynn assumes would be evident to any reasonable adult is not the same as what Michelle would assume, so there's a better than average chance that Michelle would eventually do something else that would upset Lynn.