Winkelman, MK Dons chairman, was instrumental in Wimbledon FC, FA Cup winners in 1988, relocating from south-west London to the Buckinghamshire city of Milton Keynes in 2003. The club was renamed MK Dons the following year.
While there were financial reasons for the unprecedented move of an English club away from its home town, angry Wimbledon FC fans vowed to save the club's history and formed AFC Wimbledon from scratch in the Combined Counties League.
After five promotions in eight years, AFC Wimbledon, a club owned by a fans' trust, are now back in the Football League, just one division below npower League One promotion challengers MK Dons.
The two clubs are set to meet for the first time on Sunday in the FA Cup second round.
Since the draw was made AFC Wimbledon officials and supporters' groups have called for the Milton Keynes "franchise", as they refer to the club, to drop the moniker Dons from their name.
But Winkelman said: "I've been very clear about this. I'm a custodian of the club and the only way our name could ever change is if our supporters demanded it.
"I take responsibility for the club getting here, but now it's here it's actually the responsibility of all of us.
"And I think it's incredibly clear, and clear in every conversation I have, that we are the MK Dons, we're going to stay the MK Dons, and in the future we're going to be the MK Dons."
Demanding MK Dons change their name is one of many angry reactions from the AFC Wimbledon camp since it became apparent they would have to play in Milton Keynes.
AFC Wimbledon chairman Erik Samuelson has said he will not acknowledge Winkelman on Sunday and he and the club's other officials will snub the chance to sit in the directors' box at stadium:mk.
AFC Wimbledon fans have threatened to boycott the game while some that are making the 50-plus mile trip north out of London are thought to be turning up in contamination suits.
Winkelman added: "I do understand that on this special occasion, this historic occasion, this first meeting, that they (AFC Wimbledon officials) would rather sit with their own supporters. If there is a replay, I will do the same.
"But I was disappointed to hear about the boycott, I hope that doesn't happen. Boycotting your team or leaving your team alone is perhaps the worst way of voicing your disapproval of what went on."
Samuelson has urged those AFC Wimbledon fans that do make the journey to be on their best behaviour and do nothing to harm the reputation of the club.
He told the club's official website: "We should use this as an opportunity to remind everyone just what a brilliant thing we've done over the past 10 years.
"A club's reputation is critical to its success, and our reputation comes in no small part from us always seeking to do things the right way, both on and off the pitch.
"Because our reputation is so important to us, it is critical that we conduct ourselves well while we are on our way to the game and at the game.
"There will be provocations, some deliberate but many unintended. If we react with behaviour that puts our wonderful club in a bad light, the price could be very high.
"So I appeal to all our fans not to react to provocation and, so far as is possible, to make the game a celebration of what we've achieved over the past 10 years."