Strong amalgamation property

Remarks

Clause 1 alone is the amalgamation property, so the strong amalgamation property implies the amalgamation property. Clause 2 says that an element of $B$ and an element of $C$ have the same image in $D$ only if both come from the same element of $A$.

Example: the class of groups has the strong amalgamation property. Take $D = B *_A C$, the free product of $B$ and $C$ amalgamated over $A$. By the normal form theorem for amalgamated free products, $B$ and $C$ embed in $D$ and their images meet exactly in the image of $A$.

Non-example: the class of fields has the amalgamation property but fails clause 2. Let $A = \Q$ and $B = C = \Q(\sqrt{2})$, with $f = g$ the inclusion. In any field $D$ with embeddings $f', g'$ that agree on $\Q$, both $f'(\sqrt{2})$ and $g'(\sqrt{2})$ are roots of $x^2 - 2$, so $g'(\sqrt{2}) = \pm f'(\sqrt{2})$. Hence $g'(\sqrt{2})$ lies in $f'(B) \cap g'(C)$ but not in the image of $\Q$.