What Liam said could have been true when he was a housemate, but since then, the show has changed so much, not even the producers know what the rules are these days.
Jason would have had the money for sure, it was a task he won it in, and it was being taken from the prize money, even if no one pressed the buzzer, that was said at the start.
Liam's BB money had different clauses to it.
I think it more likely that they told him he is doing well in the polls. I think it would have to be something along those lines to cheer him up after the unfair roasting he got from BB.
Well none of it should have mattered anyway because according to Jason he repeatedly says he didn't do it for the money so his will to leave should be greater than the need to keep the money, right?!! Can't have it both ways!!