I do think that cost asymmetry is a defining feature of signalling. To me, signalling is a way of getting around the problem of cheap talk. To me, a "cheap signal" is like an "unenforceable pre-commitment". It defeats the point. (Of course, many people talk about pre-commitments without actually discussing the mechanics of enforcement. I view this as a grievous omission.)

I probably was too absolutist in my criteria, they should probably be read with an invisible "ceteris paribus" attached to them. I'm happy to talk of weak and strong signals.

I want to keep 4. because I make the assumption that the audience does not be deceived. Employers do not wish to hire lazy workers, and it's in every worker's interest to say that they aren't lazy.

Regarding your birthday card example, I'd classify that as a white lie. Your coworker probably doesn't care too much if he's genuinely liked or not. Same thing with Republicanism. We could also call it a "cover story".

Re: Good managerness, what you're talking about is not signalling, but Gresham's law. Decisiveness is meant as a proxy for good-managerness. Of course good manager-ness isn't observable, that's why we would tempted to invoke signalling in the first place! "How can I show that I'm a good manager? I know, I'll act decisively!".

I do agree I was being too absolutist, I do not agree that I should modify the theory. It seems to me that once we do that, we're no longer talking about signalling as it was originally conceived. I don't know how to argue for that other than to gesture at the economics literature, which talks about deceitful employees and employers without the werewithal to sort the wheat from the chaff.

Annendum: Katja Grace discusses signalling here and uses "costly signal" and "signal" as synonymous with my version of "signal". (The previous version of this comment falsely attributed it to Robin Hanson, mental note: Always check the byline)