Mathematicians do not limit themselves in this way. Also, those who use geometry in applications, do not feel restricted to a single formal system. It is combining geometry with algebra, number systems, probability, and calculus. It is applying geometry to the world of experiences. It is using physical models, drawings, images in the imagination.
Most professors know little geometry.
In most countries there are few university-level courses in geometry and very few mathematicians who research specialty is geometry. This seems to be a phenomenon only of this century. At the end of the last century mathematicians explicitly attempted to rigorize and formalize all of mathematics by weeding out reliances on geometry and geometry intuition. It is going to be changed by human beings coming to the university who are excited by geometry and by alive geometric reasoning. In my experience, geometry is usually not most alive in the Department of Mathematics.