How does monoethanolamine (MEA) increase pH?
It is the latter case where "the whole MEA molecule is taking a hydrogen from $\ce{H2O}$ leaving the $\ce{OH-}$". Alcohols very seldom acty as arrhenius bases adding $\ce{OH-}$ to solution. One exception is triphenylmethanol but I cannot think of a second that isn't a variation of the given example. Instead the unbonded electron pair of the amine will abstract a hydrogen from water to form an ammonium ion leaving a hydroxide counter ion and thus raising the pH.