An "involved father" means a dad who actively participates in his children's lives, spending quality time with them, taking care of their needs, and being supportive and nurturing. Full definition
The breastfeeding team: the role of involved fathers in the breastfeeding family. (verywellfamily.com)
First, while children with involved fathers do much better statistically, many children who live with one parent grow up well - adjusted, and some who live with both do not. (whitehouseboysmen.org)
More involved fathers become more open emotionally, which makes them more interesting and attractive to their wives. (fathers.com)