How is Emma in both 'X-Men Origins: Wolverine' and 'X-Men: First Class'?

This confused a lot of people when Emma Frost was announced in "X-Men: First Class." In the X-Men Movie's Facebook campaign where cast members answered fan questions, Bryan Singer said they did their best to remain in continuity with the other movies but took some liberties:

Was it a conscious decision by the team to pick and choose which parts of previous movies would be canon to this movie? (White Queen already appearing in other movies, etc.)
Bryan Singer: We did our best to maintain continuity with the other films. But as sometimes happens for the sake of telling the most compelling story possible, we made a few concessions.

Then who is the "Emma Frost" in "X-Men Origins: Wolverine"?

Although this character was initially advertised as Emma Frost during a TV promo, she was credited as Kayla's Sister/Emma (Tahynna Tozzi). So, at the time, she was considered as Emma Frost - especially with her diamond powers - but since liberties were taken, she is no longer the Emma Frost viewers are familiar with.
The Emma Frost that we know is now the "X-Men: First Class" version, portrayed by January Jones. So, in a sense, this was a movie retcon.
Update! "X-Men: First Class" producer Lauren Shuler Donner also cleared up the confusion regarding the two Emma Frosts.

Is January Jones’ Emma Frost somehow the same character we saw in X-Men Origins: Wolverine?
Donner: No, she’s Emma from First Class. That’s who she was and she’s the real Emma Frost. I think January did a pretty good job of portraying her.
So there is no connection between the First Class Emma Frost and the one we see in X-Men Origins: Wolverine?
Donner: No … not really, no.