How was Vegeta able to be revived by the Dragon Balls?
I'm not sure if I understand your question, but are you asking why he was revived because he was supposed to be erased? They explicitly mention in the series that Enma Daio Sama decided not to erase Vegeta's soul just in case he needed him for a battle. So what Vegeta said in the screencapture you posted didnt apply to him due to an Enma Daio Sama decision
Enma Daio Sama decides to keep Vegeta soul intact in case he was needed for a battle
(Sorry, I couldnt find an english version for this video, only a spanish one, but you can use the english subtitles feature of Youtube to add them)