Okay, I don’t think people would call purple bacteria algae, it is a bit more complex, algae are basically cyanobacteria and eukaryotes with plastids except land plants.

I mean, all eukaryotes are Asgardarchaeota ¯\(ツ)/¯

Single-celled is physiological description, though. But I like OPs point. We have a lot of “some group except” categories. Algae (photosynthesizers except land plants), fish (vertebrates except tetrapods), some definitions of protists (eukaryotes except animals, land plants, and fungi). Practically, we use dinosaurs for the long (>65mya) extinct lineages of the clade that also includes birds. But we don’t have another name for the clade, so the point is lost.