Thanks @Animisha for elaborating. I do hear what is being said, that slurs aren't necessarily phrases, and that slurs can be components of phrases. And to be fair, the Alfred's book does say Slurs often divide the music into phrases. And above that in red it clearly says it means play legato.
Regarding the E.K.N. piece: In M10-M11the 2 slurs aren't connected. I assume that's because of the repeating C note*, but still sounds like a musical thought when it's played all together. So I understand M12-M13 is a phrase composed of slur+slur+repeating note. But what about the slurs in M12-M13 not being connected? That's the place where the Lets Play Piano Methods teacher said "they want a little separation" between those slurs (which he disagreed with).
*Although, flipping through the book I see slurs with repeating notes under them.