supporters understand the words at face value

I don't think we know this is true.

OK, that's fair. It's possible that the sympathetic hear the alleged dog-whistle but deny it, although I still think our default assumption should be to believe the supporters unless we have specific evidence otherwise. But we are still left with the puzzle of why opponents can hear the allegedly inaudible whistle.

What it predicts is that these claims should have an innocent face-value meaning but also be understandable by their intended audience as saying something else that the speaker doesn't want to say explicitly; that's all.

No. Let's look at what wikipedia says:

Dog-whistle politics is political messaging employing coded language that appears to mean one thing to the general population but has an additional, different or more specific resonance for a targeted subgroup.

You're now trying to water down dog-whistling to a mere 'plausible deniablity.' These are two distinct theories:

• When I say 'aqueducts are bad' most people think I'm arguing about the government's aqueduct-building. But members of the Anti-Bristol Society understand this to mean that I really want to persecute Bristolians, and so will vote for me. (Dog-whistling).
• When I say 'aqueducts are bad' everyone understands that I really mean that I want to persecute Bristolians, and that I'm reaching out to the Anti-Bristol Society. But because I didn't say I want to persecute Bristolians, I have just enough plausible deniability to not get thrown out of polite society. (Plausible deniability).

Note that these two ideas are essentially opposites.

When you say "In reality" and "the real meaning [...] is", are you claiming that the phenomenon described on (e.g.) that Wikipedia page never, or scarcely ever, actually happens?

Scarcely ever. Consider the example from the UK - how on earth is that dog-whistling? Both the Conservative manifesto at the last election and the official policy of the Coalition government explicitly state the goal of limiting immigration. These policies have been criticised as, inter alia, racist. So they run adverts arguing that "It's not racist to impose limits on immigration." Where's the dog whistle? Isn't the simpler story that they're trying to defend their stated policies? If this is a dog-whistle, what is the content of the dog-whistle, over and above their policies? What's the payoff? It doesn't really make sense.

Regarding your specific examples:

• I have no idea how those specific organisations, with which I have no particular familiarity, react to an individual phrase.
• Republicans certainly intend "family values" to sound more warm-and-fuzzy and positive than "heteronormativity." But there is a world of difference between using the most positive language to describe your own position and the most negative to describe the opposition (pro-choice vs pro-death, death tax vs estate tax, Obamacare vs Affordable Care Act, etc) and dog-whistling, or even plausible deniability. Social conservatives are not hiding anything by saying they favour "family values." They are perfectly willing to defend each and every one of the policy planks, but they want one phrase to sum it all up.
• I'll never be able to prove that nothing they did was intended like that. But I certainly don't believe that the most-remarked-on incident (Bill Clinton comparing Obama's win in the South Carolina primary to Jesse Jackson's win in the South Carolina primary) was in any way a dog-whistle, and I certainly do believe that what really happened there was that an innocuous comment was seized upon by people as a stick with which to beat Clinton, with the convenient excuse that the very innocuousness of the comment is the evidence for its malignity, because the evil is somehow hidden.