Aside from being a silly quibble (more on that below), it is quite selective with the truth. Yes, the flag on the right is the "first Confederate National Flag," and no, it does not exhibit the "Southern cross."
But what about the second Confederate National Flag (1863-1865)? Or the third one (1865)? Here they are:
The Second Confederate National Flag, a.k.a. the "Stainless Banner."
Look closely at this flag. Do you notice anything that distinguishes it from a standard-issue, plain-vanilla flag of truce?
The Third Confederate National Flag, a.k.a. the "Bloodstained Banner."
In fact the Second Flag was sometimes mistaken for a flag of truce, so the Third Flag added a distinct element: a red stripe along the edge. Lacking any other distinguishing feature, that could mean that the Confederacy had developed a thirst for Jamaican beer.
But there is of course a distinguishing feature on both of these Confederate National Flags: the Southern Cross.
So the "fact" as reported is less than honest. It is also a silly quibble. The Confederate Battle Flag is the most important emblem of the Confederacy, because it was the flag that most of its soldiers fought and died under. Even if it wasn't the flag of the government, it was for them the flag of The Cause. And what was that Cause? It was to found a nation in which "no bill of attainder, ex post facto law, or law denying or impairing the right to property in negro slaves shall be passed" and in which no dang-blasted Yankee states would be able to practice states' rights by refusing to send fugitive slaves back to their owners.
And if you don't believe me, you can read the Confederate Constitution.