After just one disappointing campaign, immediately after his free transfer from Chelsea in the summer of 2010, the 30-year-old was loaned out to Ligue 1 side Lille.
There he enjoyed a return to form, scoring four times in 27 appearances and rediscovering his creative side.
Now he is back at the club's Melwood training ground for pre-season looking for a fresh start under new manager Brendan Rodgers, whom he knows from their time together at Stamford Bridge.
"I don't want to leave Liverpool one day having not been a success here or having left something here," said Cole.
"I want to make an imprint at the club. I've been a success at West Ham, Chelsea and Lille - I don't want people to be able to say that I haven't here.
"I'd like to be given the opportunity to play, learn and improve, and join this great team.
"It's a great group and I would like to be involved in this team and really push on."
The midfielder has played 32 times for the Reds but made just nine Premier League starts.
However, he believes his experience in France last season, coupled with a determination to be better, will serve him well second time around - especially with Rodgers now in charge after previous managers Roy Hodgson and Kenny Dalglish failed to get the best out of him.
"It's a clean slate for everyone, so I'm delighted to be back," he told liverpoolfc.com.
"You have to remember, I've only been at Liverpool one season, really, but this is the third manager I've played for.
"It does feel different this time - it feels like it's ready to go.
"I came here for one reason - because I wanted to be a part of the resurgence of this club.
"It excited me coming here because I knew what a big club it was with a rich history and I wanted to be part of something. I still do.
"The fire still burns in me to be a part of it - I want to be part of this club moving in the right direction.
"When I came here I was sent off in my first game and had a few little niggly injury problems.
"I think I only played something like 700-odd minutes of Premier League football in that season.
"It was tough for me, far and away the worst season in my career because I didn't get going.
"I'd like to improve on that. I'd like to be a part of this great club, this team and everything about it."
Cole said playing in a foreign country for the first time had given him a fresh insight into the game and he wanted to bring that to Liverpool.
"I loved the football and we had a successful season," he said, having helped Lille secure Champions League football with a third-place finish.
"I'd always wanted to play abroad and sometimes you think the chance to do so at a good level may have passed by, but it was great and I really enjoyed the experience.
"I love my football, I love training and I love learning - and I did learn a lot out there.
"It opened my eyes to a new way of approaching football and the continental way.
"I'm always trying to learn, pick up new things and improve myself. I strive to get better every day."