People on my side of the issue want skepticism to be about appreciation and enjoyment of science, and critical thinking, which of course is not philosophy-specific. After all, if your concern is showing people scientifically why homeopathy is quackery, religion isn’t an issue. And skeptics are more concerned with things like homeopathy than with the meaning of life. This doesn’t prohibit us from evaluating/criticizing fact claims made in the name of religion if such claims are shown to be wrong through science. We can show through science why the catastrophic cutting of Grand Canyon by a Flood is wrong, and do. On the other hand, if science is about explaining through natural causes, it’s not a scientific question whether God miraculously raised Jesus from the dead.
The distinction also has to do with one’s definition of science, and whether that definition includes philosophical as well as methodological naturalism. Mine doesn’t. We don’t want to say “sorry, you can’t be a critical thinker and enjoy science unless you are a nonbeliever” because that is poor philosophy of science, and additionally would unnecessarily truncate our movement.
So BAS is an “old fashioned” skeptical organization that is religiously-neutral. Most people have more than one identity: I’ll wear my humanist hat in some circles, but not at the bee-keeper’s meeting. One can also be a BAS member and not have to wear an atheist hat to do so.