The planet is not just a rock in space. It includes the environment and everything that is there. When we destroy that, we are destroying the planet itself.
I think by focusing in on that line you're missing the bigger picture. What I'm trying to say is that we're primarily a threat to ourselves, not to the 'planet'.
I think people would be much more engaged if it weren't "save the planet" but rather "save the human race".
Except... not? Altering, or even completely destroying the biosphere is not nearly the same thing as destroying the planet. Mass extinction events have occurred in the past, in terms of history on a planetary scale the anthropecene's mass extinction is nothing new.
Extinctions happened were natural, nobody could stop it. This time it's because of greed and selfishness. It could have been stopped but not and still going on even though we know it. Don't justify this time with before.
Also we are not debating whether that happened before or not. Some (mostly american from what I've seen) people will take any excuse to justify it.
I include my hair, but if I shave my head I haven't destroyed myself.
What is really going on is that “We’re killing the planet” is a hyperbolic way to express “We are destroying the things I personally value about the planet” or perhaps “We are making the planet unsuitable for ourselves”; trying to defend it as something else is not only usually wrong on its own terms, but missing the point of what people actually mean when they say it.
The planet WILL be fine, the solar system, universe, existence.. Yea, in the big scheme of things it WILL be fine. Conservation is about conservation of the human species, and there is no reason we shouldn't talk about it like that.
Honestly, more right wing / anti-environmentalists would probably respond better if they understood how dire the threat to humanity is.