That's magic for you, tangled and intractable rules and non intuitive exceptions. I say this positively. It checks out, power of names, you are your name, if it's said, then implicitly some part of you is now inside. Implicite summoning.
What's more odd is that this particular rule was never mentioned prior to this point, and if the characters were aware, they should have been actively trying not to say certain names. These are professionals, they shouldn't be getting tricked by seeing an illusion and involuntary saying who they see.
It feels like an idiot ball.