Debate I had with my friend about the definition of a religion

I argued that while Christianity and Islam are two different religions, Catholicism, Orthodoxy, Protestantism are different branches of the same religion. My reasoning is that the three Christian branches share the same core doctrine (trinity, salvation through the cross, etc.) but just differ on practice and rituals whereas Christianity and Islam have fundamentally different core doctrines (Trinity, crucifixion vs Tawhid, Jesus being replaced and sent directly to heaven). Of course my underlying point is that sharing core doctrines is what differentiates a branch of a religion from a religion.

My friend argued that the different Christian branches are all different religions in the same way Christianity and Islam are different religions. He thinks that not only core doctrines, but also practice and rituals are core to the distinction. So Catholicism and Protestantism would be different religions because of papal infallibility vs not, faith plus works vs sola scriptura, etc. But I find this nonsensical though because then my childhood denomination (the United Church of Christ) would fundamentally be a different religion from United Methodists or Presbyterians, which I find ridiculous. Who do you agree with and why?

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from Islam https://ift.tt/buceCrG