There are isomorphic Coxeter groups with different Coxeter diagrams. So a simple answer to your question is "no". Nevertheless, the sets of isomorphism classes of Coxeter groups given by Coxeter diagrams are not very large, and that information can be viewed as the "almost yes" answer to your question. See, for example Mihalik, Michael, Ratcliffe, John, Tschantz, Steven, Quotient isomorphism invariants of a finitely generated Coxeter group. Aspects of infinite groups, 212–227, Algebra Discrete Math., 1, World Sci. Publ., Hackensack, NJ, 2008 or Marquis, Timothée, Mühlherr, Bernhard, Angle-deformations in Coxeter groups.
Algebr. Geom. Topol. 8 (2008), no. 4, 2175–2208 and the references there.