On the following link, https://perso.univ-rennes1.fr/matthieu.romagny/exposes/Theoremes_de_comparaison_et_representatons_galoisiennes.pdf, page, $ 2 $, the Berthlot-Ogus isomorphism theorem is stated as follows,

We have a canonical isomorphism, $$ rho_{mathrm{cris}} : H_{mathrm{cris}}^{i} (X) otimes_{K_ {0}} K to H_{mathrm{dR}}^{i} (mathscr{X}) otimes_{O_K} K = H_{mathrm{dR}}^{i} (X) qquad (1) $$

On the other hand, on the following pdf, https://www.jmilne.org/math/Books/DMOS.pdf, page, $ 359 $, the author claims that there is an isomorphism, $$ H_ {mathrm{DH}} (X) otimes W simeq H_{mathrm{cris}} (X (s) / W) qquad (2) $$.

What is the difference between the two isomorphisms $ (1) $ and $ (2) $ ?

Are the two isomorphisms the same ?

Thanks in advance for you help.