It's fine to get excited about strongly relativistic arguments but I don't think they have a place in talking about events at distant celestial objects.
The amount of distortion you can have between distances and times is bounded by the amount you let your reference frames differ. Between stars there is not all that much velocity, so time is a relatively stable concept. Sure it'll wander a few percent but nothing nearly as drastic as having no sense of 'now'.
Don't forget that when you bring relativistic speeds into the mix you can't agree on distances either. Are you prepared to argue that it's actually one billion light years away?
When you read someone write "as I write this", they mean "simultaneous with the event of this writing in the reference frame that I occupy currently". I mean I doubt anyone conceptualizes it like that, but that's what they mean physically. It just wouldn't make sense any other way.
So what does it mean when someone writes "as I write this, the water is all gone"? That statement is factually false, since in their frame of reference the event in which the water disappears has not yet happened.
Now as I noted in my thought experiment, that statement is true in "a frame of reference which agrees with the theoretical frame of reference of a spaceship we launch today, at the time it arrives at the reservoir". But that's just not what people mean when they say "now".
Commenters on this article, and there are several, aren't saying that because they're mindful of relativity and are using a peculiar implicit reference frame in order to make the point that the water might all be gone by the time our ship gets there. If they were, they would never say "now", because that's nonsensical, and we have a perfectly straightforward way to say that: "It might all be gone by the time our ship gets there." (In fact, isn't "the frame of reference of a spaceship we launch today, at the time it arrives at the reservoir" the same as "the time our ship gets there"?)
They're saying it because they think that there's such a thing as simultaneity. They think conceptually we could travel there instantly. It's just not true.
And it's not not true in the sense that "little g is 10m/s^2" is not true, it's not true in the sense that "The sky is red" is not true. Somebody who says the sky is red misunderstands something fundamental about their universe, and I think that's worth teaching them about.
Now, sure, you could argue that, say, it's stormy out, so it's dangerous to fly, just like it would would be if the sky were red, so practically the statement is ok and I'm just being a pedant. But that would be ridiculous.
But that sort of problem only comes up when you accelerate. If you maintain a frame of reference then you can handle things far more gracefully.
A shared reference frame over two points billions of light years apart can be established. You simply keep the distance between these points roughly constant, which holds the velocity of all points roughly constant, which keeps calculations of simultaneity roughly constant across the whole region.
I can then state that in the combined earth/reservoir plusorminus .01c reference frame that the water is already gone. And this is factually true. You still have to wait for information to propagate but everything works out. The timing has no disputes between any members subscribing to this reference frame.
Relativity doesn't cause major problems until you deal with matter moving at high relative speeds.
P.S. I'm deliberately ignoring the expansion of space in this argument.
The amount of distortion you can have between distances and times is bounded by the amount you let your reference frames differ. Between stars there is not all that much velocity, so time is a relatively stable concept. Sure it'll wander a few percent but nothing nearly as drastic as having no sense of 'now'.
Don't forget that when you bring relativistic speeds into the mix you can't agree on distances either. Are you prepared to argue that it's actually one billion light years away?