Isomorphic quartic K3 surfaces in the view of Cremona and projective transformations

We show that there is a pair of smooth complex quartic K3 surfaces $S_1$ and $S_2$ in ${\mathbf{P}}^3$ such that $S_1$ and $S_2$ are isomorphic as abstract varieties but not Cremona isomorphic. We also show, in a geometrically explicit way, that there is a pair of smooth complex quartic K3 surfaces $S_1$ and $S_2$ in ${\mathbf{P}}^3$ such that $S_1$ and $S_2$ are Cremona isomorphic, but not projectively isomorphic. This work is much motivated by several e-mails from Professors Tuyen Truong and J'anos Koll'ar.