Many hospitals have a clear religious purpose, yet that does not make them or the medicine in them religion.
Muslims aren't trying to get ID in schools as science.
Oh?
Most Christians are able to separate their personal religious belief from science. Others apparently think their belief should take the place of science.
Which makes Christians distinct from ID'ers by your definitions(Since not all ID'ers are Christians and not all Christians are ID'ers). Therefore your step 3 does not apply to ID'ers.
It has been clearly equated by the founders and leaders of the ID movement in this country.
The Muslim I quoted above doesn't think so.
One of your problems is that you are equating the Discovery Institute with ID. They are not. No more than NCSE is Darwinism.
Hospitals have helping people as their mission. ID has spreading Christianity and changing culture as its mission.
And it looks like Muslims are riding on the coattails of the Christian ID work.
Which makes Christians distinct from ID'ers by your definitions(Since not all ID'ers are Christians and not all Christians are ID'ers).
The ID movement in this country was started by Christians for Christian purposes. That all Christians aren't in it, or that others believe too is absolutely irrelevant.
You can argue opinion over this, but the fact remains that the logic was solid, it was not a fallacy.
One of your problems is that you are equating the Discovery Institute with ID.
The Discovery Institute founded the ID movement in this country, set its strategies and provides funding. "Teach the Controversy" and the idea of evolution as a theory in crisis and dissent is their invention. I haven't found one of the top-rung ID proponents who doesn't have some connection to the Discovery Institute (Behe, Dembski, Berlinski and Gonzalez from the movie are actually fellows).