If I could only have one, I'd choose the 70-200L II. Bare lens, its AF is better than the 70-300L and 100-400L. With a 1.4x, it's IQ is similar to the 70-300L and it's a stop faster (f/4 at 280mm vs. f/5.6 at 300mm) than the 100-300L. Better for portraits, sports, etc. Yes, it weighs more, costs more and is physically longer than the 70-300L (in locked position) but it is that good.
However, if you plan on using it outside and/or for travel, then the 70-300L is also very good choice. It stores more compactly and is less awkward to handle than the 70-200L II + 1.4x, but losing at least a stop hurts when shooting sports/shows indoors.
If you can, try them both in a store and see if the weight/handling difference matters to you. I use the 70-300L when visiting places like the zoos or for daytrips. For everything else, the 70-200L II is the choice.