I would like to push back against accepting gaslighting if it doesn't appear to materially affect you at the time. You make yourself vulnerable to being gaslit about how materially affecting it can be to you. This is how abusers destroy the identity of their victims over time, through continually downplaying their harm and convincing the victim they are not being injured. This makes it extremely hard for someone to leave their abuser, because their psychological state has been warped that being abused is the norm and therefore there's nothing better than the abuse out there.
If Alex is willing to lie and also gaslight someone about breakfast, Alex isn't safe to be around. Especially for someone who has been previously abused.
I don't think anyone was suggesting you should still be fine with Alex if you caught him manipulating you, only that it's silly and pointless to be immediately distrusting and demand a bunch of evidence when someone makes a claim that doesn't matter. That is, be generally trusting of Alex (and everyone else) as long as the stakes are low.
That wasn't clear from your original post, sorry, it read more to me that there is a level of gaslighting that is generally acceptable. I think even if the stakes are low, someone who lies and gaslights over low stakes is no longer a low stakes scenario.
Lying and gaslighting are two different things. Gaslighting involves lying but is also about invalidating the victim's memory and feelings in an attempt to control them in some way. Telling a partner they look good is not generally gaslighting.
Lying isn't the same thing. In that moment, the man is just being dishonest. Later, if the wife brings up the incident and the man claims it both never happened and she constantly makes shit up like this, then yes he is gaslighting and is deeply unsafe.
If Alex is willing to lie and also gaslight someone about breakfast, Alex isn't safe to be around. Especially for someone who has been previously abused.