If I were you, considering that:
you already have a 6D,
that wildlife will just be a part of your photography,
that long lens work is something that is new for you,
and that $2,000 is your maximum budget . . .
Buy a 50D, used, for around $400.
Buy a 400 f5.6 (I just bought one that was absolutely perfect for $950, all inclusive).
Spend the remainder of the $2,000 - about $600 - on a very sturdy tripod, a heavy-duty ballhead, and a quick release plate.
This setup will give you a field of view equivalent to 640mm...
Different horses for different courses.
I would consider a used crop camera, say a 50D for around $400, or whatever good deal you can find.
Then the Tamron 150-600, which is only 1/3-stop slower than the 400 at the 600mm end (i.e., when it is 200mm longer), and has image stabilization and a 6-year warranty, for $1069 new.
I would also consider the tripod advice, but I almost never use a tripod when birding. But you might, and you might also find some other uses for a long lens where a good tripod would help. (Without the tripod you could get a good used 7D and have a much better wildlife camera.)
This would give you a field of view equivalent to 960mm.
Like many here, I am interested in how the Sigma 150-600 Sport turns out, but it would consume your entire budget, and with the 6D your FOV would be 600mm. Not bad, but not 960mm. The Sigma 150-600 Contemporary is likely to be much like the Tamron, but at this point it is a vapor product.
Jake is absolutely correct about both the 400 and the 100-400. You have to balance several parameters in making this decision: Length (which in these scenarios includes the effect of a crop sensor), price, focusing performance (which a lens does not do by itself, but with the AF system of the camera it is attached to) IQ, weight, etc. For me, the "reach" of the 150-600 lens outweighs its weakest point, which is its focus speed. You have to decide what is most important you you.