I'm well aware of what the tutorial says.
So again, no, they are not linked. Both Energy and wakefulness (what I call it; maybe fatigue is a better word) are affected by walking around and doing things. While Energy can be increased by using potions and by sleeping, wakefulness can only be increased by sleeping. So you can't just go around the game using energy potions all the time without sleeping; at some point, you have to sleep.
But while losing Energy does not cause you to receive debuffs during combat, becoming tired (by the three degrees of fatigue) does cause debuffs. There is no relation between the two -- being tired doesn't cause your Energy to drop any faster, or suddenly get lower; having low Energy doesn't mean the MC is tired; and having a lot of Energy doesn't make you any less tired--even if you have max Energy, if you are too tired then you cannot do actions. The only common element is that they are both affected over time, and can both be resolved by sleeping.