The go-to secondary schools that parents have liked much better (and which are not Int'l) are the Elisabeth-von-Thadden Schule and St Raphael.
... and both are private schools, like EI and, with far worse reputation, Heidelberg College and, outside the general competition, the Waldorf school. Sure, among those four german private schools in Heidelberg, those two are the go-to. In competition with the state schools? Part of the second string, with Helmholtz, Bunsen and Hölderlin having a similar reputation, and KFG usually rated higher (but that might have to do with the fact that you have no chance there unless your parents hold at least a Prof title or donate certain amounts of money to the school).
Each school in Heidelberg pretty much has its own reputation. Not so much in comparison to the others, but by itself.
They're Lutheran and Catholic (respectively) so maybe not everybody's taste but of course they accept pupils from all religions and from no religion at all.
And staunchly so. Thadden forces pupils to participate in social projects with Diakonie and intern there, and has regular prayer hours and mass with the classes. Raphael afaik does the social projects and internships with Caritas, but not the regular praying (at least not any more than at other schools). It used to be a all-girls school under a nun order btw, the Realschule connected to it still is one today.
Thadden is part of the school foundation of Diakonie, while Raphael is part of the school foundation of the archdiocese of Freiburg. And of course they accept pupils from all religions. Otherwise, they'd lose their school license damn quick.