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Regarding the question "Who was the person who invented this notion?", a paper of H. T. Muhly provides interesting background (as well as a geometric characterization)interpretation) for projectively normal:

In the terminology of the Italian School an algebraic variety is called "normal" if its system of hyperplane sections is complete. O. Zariski applies the term "normal" to an algebraic variety whose associated ring of homogeneous coordinates is integrally closed. The two concepts are not equivalent. Zariski refers to a variety which satisfies the former condition as "normal in the geometric sense" and to one which satisfies the latter condition as "normal in the arithmetic sense".

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The object of this note is to characterize geometrically those algebraic varieties which are normal in the arithmetic sense. To this end we propose the following theorem: A necessary and sufficient condition that the $r$-dimensional algebraic variety $V_r$ be normal in its ambient projective space $P_n$ is that for every integer $m$ the linear system cut out on $V_r$ by the hypersurfaces of order $m$ in $P_n$ be complete.

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Regarding the question "Who was the person who invented this notion?", a paper of H. T. Muhly provides interesting background (as well as a geometric characterization):

In the terminology of the Italian School an algebraic variety is called "normal" if its system of hyperplane sections is complete. O. Zariski applies the term "normal" to an algebraic variety whose associated ring of homogeneous coordinates is integrally closed. The two concepts are not equivalent. Zariski refers to a variety which satisfies the former condition as "normal in the geometric sense" and to one which satisfies the latter condition as "normal in the arithmetic sense".

(...)

The object of this note is to characterize geometrically those algebraic varieties which are normal in the arithmetic sense. To this end we propose the following theorem: A necessary and sufficient condition that the $r$-dimensional algebraic variety $V_r$ be normal in its ambient projective space $P_n$ is that for every integer $m$ the linear system cut out on $V_r$ by the hypersurfaces of order $m$ in $P_n$ be complete.