The 23-year-old midfielder joined Villa on a season-long loan last year with an obligation to buy conditional on appearances, but Villa opted against activating the clause after limiting his involvement under manager Unai Emery.
The arrangement would have become permanent if Elliott reached a specified number of appearances, a threshold he never met.
Elliott, who has featured in Liverpool's first two pre-season matches under new manager Andoni Iraola, said his return to Anfield felt like a chance to reset.
"I'm here and it's kind of a second chance, if you want to call it," Elliott told reporters on Thursday.
"Especially with the gaffer and new coaching staff coming in, it's another chance to impress and hopefully get in their good books.
"I wish Villa nothing but the very best... Unfortunately, the situation happened but it's another learning curve. I'm here now and I'm trying to cement my spot in the team or within the squad."
Elliott, who joined Liverpool in 2019, finished the last season with only nine appearances across competitions for Villa and said the experience was among the toughest challenges of his career.
"Mentally, it was a very tough one, it was a learning curve," said Elliott, who was part of Villa's Europa League-winning campaign.
"It was an opportunity for me to experience something new, as much as I didn't want to do it, and I don't wish it upon any other player, because it's not nice at all.
"But I have no hard feelings. I'm on the other side of it now. I was able to get through it and achieve something at the end of the season as well, another historical moment for the club, and to be part of that was another great situation."
Liverpool, who finished fifth last term, begin their Premier League campaign at Newcastle United on August 23rd.