Why is this organic compound trans, and not cis? I'm reading a high school chemistry course online, and I'm now reading about geometric isomers.

They provide this example:


And in terms of naming it, they say this: 


  Begin by identifying the parent chain - it is the continuous chain that contains the carbon-carbon double bond. The two groups that belong to the parent chain are on opposite sides.


Then supply this illustration: 

And with that explains that it's trans-3-ethyl-3-hexene for that reason. I don't quite get their explanation though, becuase to me it seems like this could also be cis.

Why can't it be this one: 

As far as I understand, that's also a valid way of expanding it, because it has the same amount of carbons in the chain and the numbers are still as low as they get. The only difference is that this one would be cis-3-ethyl-3-hexene instead of trans-3-ethyl-3-hexene. 

So, why exactly is it that only one of them is right?
You are right, there is no difference between the two molecules that result if you rotate 180 degrees about the $\ce{C=C}$ double bond - they are the same exact molecule.  So for this molecule, the terms "cis" and "trans" would not be used, they have no meaning here.  The name of the molecule is simply 3-ethyl-3-hexene.  If the on-line course says otherwise, it is wrong.