The notion of homomorphism-homogeneity, introduced by Cameron and Nešetřil, originated as a variation on ultrahomogeneity. By fixing the type of finite homomorphism and global extension, several homogeneity classes, calledmorphism extension classes, can be defined. These classes are studied for various languages and axiom sets. Hartman, Hubička and Mašulović showed for finite undirected L-colored graphs without loops, where colors for vertices and edges are chosen from a partially ordered set L, that when L is a linear order, the classes HH and MH of L-colored graphs coincide, contributing thus to a question of Cameron and Nešetřil. They also showed that the same is true for vertex-uniform finite L-colored graphs when L is a diamond. In this work, we extend their results to countably infinite L-colored graphs, proving that the classes MH and HH coincide if and only if L is a linear order