Perhaps the Noda Beyehuda has other Rishonim, not like that Tosafos. Come to think of it, the Gemara (including Chesronos Hashas) mentions Yom Alef Shel Notzri along with Yom Eidam of Avodos Zara.
mdd, they have other problems, too. There are icons. Also, Apikursus is also Yehareg V’al Yaavor, which leads me back to the original topic here, that a public acknowledgement to a different belief is like a Modeh Ba’avoda Zara.
What would you think about someone getting up in front of a Christian group and reciting a Christian prayer that happens not to use any questionable term?