Oswald in the doorway: the blog of the Oswald Innocence Campaign, by Ralph Cinque

I think it's very likely that the version of Lovelady that was sent to the Warren Commission by the FBI was the soft one on the right. 

 The Idiot Backes can't see that the shadow softens the image. But, it's neverethless true that he looks younger and gentler on the right. On the left, he looks harder, as in tougher and meaner. In a word, he looks ugly. He looks like a thug. On the right, he doesn't.

You have to remember what they were shooting for at the time, which was a guy who could easily be mistaken for Oswald. Remember the stories? That Mrs. Lovelady showed up at the TSBD and saw Oswald from a distance and called to him like he was her husband. So, she mistook a 131 pound man for her 170 pound husband. And, his step-kids supposedly saw Oswald on tv and thought he was their dad. 

So, they were trying to sell the idea that Oswald and Lovelady looked very much alike, and Oswald had a soft look. I believe that's why they flipped so many of his images: because they thought he looked more shrewd and cunning and diabolical when flipped. 

As late as 1979, the HSCA published a flipped image of him. 

 He looks older and harder flipped than not.

It's a real problem when your Mad Dog Killer is NOT a scary looking guy. But, they realized it, and that's why I say that they would not have sent a scary image of Lovelady when they had kinder/gentler one. They were trying to say: "You see: he could easily have been mistaken for Oswald." And that's why they would have gone with the Soft Lovelady. 

So, why would the Mary Ferrell Foundation be showing the wrong version on their website?

I don't know. But, that is what they show.