The genetic-environment correlation where children with different genetic composition evoke responses from adults that strengthen the child's genetic inclinations is called

The described genetic-correlation is called evocative or reactive correlation. The general idea of gene-environment correlation describes the phenomenon when one's genotype affects its environment. It involves three different types: passive, active and evocative gene-environment correlations.
An example of evocative correlation could be the case where a child shows ability in basketball, while his/her parents have no association with sports. However, they may react by increasing the child's basketball experiences. This reaction will reinforce the child's genetic inclination towards basketball.