Why are turtles classified as reptiles and not amphibians?
Amphibians are not defined for having a moist skin, neither are reptiles defined for having scales on their body.

In biology, organisms (elements) are grouped according to their evolutive history in monophyletic groups, also known as clades. Basicaly, a monophyletic group is


  A group consisting of an ancestral and all its descendants.


"Tetrapoda" is a monophyletic group, formed by vertebrates with four limbs. Inside Tetrapoda, there are two groups: "Amphibia" and "Amniota". Amphibians are non-amniotes tetrapods, and we have some doubt if they are a monophyletic group. Amniotes, on the other hand, are tetrapods having an amniotic egg, including you and me, and they form a monophyletic group.

The amniotes are divided in two monophyletic group: "Mammals", which are amniotes with a synapsid skull, and "Reptiles", which are amniotes with diapsid skulls (I'm using "reptiles" as synonym of Sauropsida).

Reptiles include turtles, lizards, snakes, alligators and dinosaurs (which include the birds: all birds are dinosaurs). It doesn't matter if a animal has or has not scales, or if it lays eggs or if it is viviparous, or if it has 5 fingers or 3 fingers: All the descendants of a given ancestor are included in the monophyletic group that contains that ancestor. 

Turtles, despite being strange or somehow different, are descendants from the same most recent commom ancestor of Reptilia... that's why turtles are reptiles (and that's why birds are reptiles as well).

To make this more clear, have a look at this cladogram (from Hickman, Integrated Principles of Zoology):



The apomorphy that defines the tetrapods is "paired limbs". You have Amphibia to the left and Amniota to the right, whose apomorphy is " egg with extraembrionic membranes". Inside them, you have Reptilia, whose apomorphies are "skull with upper and lower fenestra and beta-keratin in epidermis". Turtles came from an ancestor with these characteristics. So, turtles belong to the monophyletic group of "Reptiles".

Post scriptum: You wrote that "turtles (specifically sea turtles) live on both land and water, very much like amphibians". Just a curiosity: the reason why sea turtles leave the water (sea) from time to time shows exactly that they are not amphibians! Amphibians, being non-amniotes, have eggs that survive under water (actually, with few exceptions, they need to be under water). Turtles, on the other hand, are amniotes, and the amniotic egg cannot be laid under water. That's why the turtles have to leave the water to lay eggs: because, contrary to the amphibians, they cannot lay eggs under water.