Supposing a spaceship going by the Earth, you're figuring the spaceship has the velocity, not the Earth. Why not the other way around? In special relativity there's no way to say one is holding still, the other moving; or rather you can say either, depending on reference frame. So you would have to say they both have the same time dilation, so they're equal after all ...
That's why the detail of bringing them back together, so you can compare them, is vital. If the spaceship turns around and comes back, it will show dilated time. On the other hand if the Earth runs after the spaceship fast enough to catch it - it will show the reduced time instead! (By SR, ignoring GR). You can imagine the spaceship sends a rocket or a light signal with a time stamp, or that the Earth does so; or that there's an intermediate space station receiving, and comparing, the signals; and so forth. If you carefully take everything into account (calculating with actual speeds and distances in your thought experiments) you'll find SR always comes through with a sensible, consistent answer.
Given your doubts, this is a very worthwhile exercise.
Now the "Einstein online" reference you mention is, most likely, claiming we can know that the spaceship is the one which actually experiences time dilation. So it must assume the existence of an absolute reference frame, such as the microwave background radiation. And, that the Earth is almost at rest with respect to it, compared to the spaceship at very high velocity.
I don't say the article is wrong, especially since you haven't given a reference so I can look at it. But it's probably no good. If it presented a real proof, it would overturn 100 years of accepted physics, so we should have heard about it. Nevertheless post a ref and no doubt people will take a look.