The following paper gives an example showing why distributive lattices have amalgamation but not strong amalgamation.

E Fried, G Grätzer. Strong amalgamation of distributive lattices. Journal of Algebra, Volume 128, Issue 2, 1 February 1990, Pages 446–455. doi:10.1016/0021-8693(90)90033-K

Their example is the following.

Let A and B be the 4-element Boolean algebras A = {s0,s1,s2,a} and B = {s0,s1,s2,b} with s0 bottom and s1 top. Let S = {s0,s1,s2} be the intersection of A and B. S is a distributive lattice, indeed a sublattice of both A and B. Since both a and b are complements of s1, the 4 element Boolean algebra D = {s0,s1,s2,c} where c identifies a and b, provides an amalgamation of A and B wrt S. But S is not the intersection (formally: pullback) of A and B in D (as both A->D and B->D are isomorphisms).

The following paper gives an example showing why distributive lattices have amalgamation but not strong amalgamation.

E Fried, G Grätzer. Strong amalgamation of distributive lattices. Journal of Algebra, Volume 128, Issue 2, 1 February 1990, Pages 446–455. doi:10.1016/0021-8693(90)90033-K

Their example is the following.

Let A and B be the 4-element Boolean algebras A = {s0,s1,s2,a} and B = {s0,s1,s2,b} with s0 bottom and s1 top. Let S = {s0,s1,s2} be the intersection of A and B. S is a distributive lattice, indeed a sublattice of both A and B. Since both a and b are complements of s1, the 4 element Boolean algebra D = {s0,s1,s2,c} where c identifies a and b, provides an amalgamation of A and B wrt S. But S is not the intersection (formally: pullback) of A and B in D (as both A->D and B->D are isomorphisms), therefore D is not a strong amalgamation. Moreover, it follows from D being the pushout of (S->A,S-B) that no other amalgamation of A and B wrt S is strong.

The following paper gives an example showing why distributive lattices have amalgamation but not strong amalgamation.

E Fried, G Grätzer. Strong amalgamation of distributive lattices. Journal of Algebra, Volume 128, Issue 2, 1 February 1990, Pages 446–455. doi:10.1016/0021-8693(90)90033-K

The following paper gives an example showing why distributive lattices have amalgamation but not strong amalgamation.

E Fried, G Grätzer. Strong amalgamation of distributive lattices. Journal of Algebra, Volume 128, Issue 2, 1 February 1990, Pages 446–455. doi:10.1016/0021-8693(90)90033-K

Their example is the following.

Let A and B be the 4-element Boolean algebras A = {s0,s1,s2,a} and B = {s0,s1,s2,b} with s0 bottom and s1 top. Let S = {s0,s1,s2} be the intersection of A and B. S is a distributive lattice, indeed a sublattice of both A and B. Since both a and b are complements of s1, the 4 element Boolean algebra D = {s0,s1,s2,c} where c identifies a and b, provides an amalgamation of A and B wrt S. But S is not the intersection (formally: pullback) of A and B in D (as both A->D and B->D are isomorphisms).