“The problem with the analogy is that, in most of these cases, the same-sex couple is not requesting a different type of cake. They are requesting the same exact cake (same shape, same frosting, same everything).”
The above does not make sense in the context of your following sentence:
“The only difference is that they plan to use the cake in a same-sex “wedding.”
If it is all the same, how could it possibly be known it is for a same sex “wedding”?
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The thing is, you can go in and on with this sort of minutia, as you are doing, but it side steps the main issue of what rights do business people have with respect to their religious or other closely held beliefs and that applies to all religions, including atheists.
The above does not make sense in the context of your following sentence:
“The only difference is that they plan to use the cake in a same-sex “wedding.”
If it is all the same, how could it possibly be known it is for a same sex “wedding”?
Because the two people getting "married" may come in to shop for the cake together. Couples often do.