Your impression that "profess" suggests something claimed to be true is not so - is mistaken. It does not have that meaning. It is possible that the thing not be true, but in that respect it acts just like the verbs "claim" and "assert". If you say "Jack claims to be an excellent writer", one might infer that Jack is not an excellent writer, but only because excellent writers do not make a habit of claiming to be excellent. It is the context, not the word "claim", that suggests the claim to be false. The same is true for "profess". The word does not suggest anything to be false. To understand Latin words like "profess", I always start by looking the etymological meaning. The meaning of Latin words changes only a little bit over time. The Latin roots mean "to bring forth a confession". From that, the word evolved to mean "to declare openly or make a public statement". That is what it means today. The Latin root of "purport" means "to carry forth". That is similar to the Latin root of "profess" but there is an important difference. Since a confession is something personal and sincere, it is far more personal and convincing than something that is merely carried forward. That difference persists to this day. Human beings profess something, but any object can purport something. For example, you can say "This restaurant purports to serve classic French cuisine, but it does not." By contrast, a restaurant cannot profess anything because it has neither a mind nor a conscience.

I would use "asserts" instead of "profess" in your examples. "She asserts this meaning [is] embedded in that word."