Can a natural stone wall be made waterproof without defacing it?
Where is the water damage relative to the exterior ground level? I note in the outside picture that there is significant slope to the ground - if the ground on the uphill and slope-side sides is not shaped to move water away from and around the house, it would be likely that there would be water damage on the lower parts of the walls from water flowing over the ground into the house - that would be corrected by reshaping the ground, not anything on the above-ground wall.

You could certainly try a clear-spray-on waterproofing material on the stones - but that won't solve ground drainage issues, if those are the problem. Digging out the soil around the house and applying waterproofing to the below-ground wall and placing a drainage pipe in the bottom of the trench before refilling will address those more throughly if reshaping the ground does not completely solve the problem.