After recent talks, Chester was on the verge of signing the deal for 2022 before his surprise exit.
Sunday’s 22-18 loss at Huddersfield Giants - when Trinity squandered an 18-0 lead - prompted a change in heart from the club’s board and assistant Willie Poching has been made interim head coach.
Chester, who was out of contract in November, had been in charge since March 2016 but Trinity struggled badly last term and a current five-game losing run once more saw them stuck second bottom in Super League.
Carter, who described it as the “toughest decision” of his reign, admitted: “We’d had some talks about it and we’d got to an understanding of agreeing an extension.
“It wasn’t the reported two-year deal that had been on social media but we had got to an understanding on it.
“But I think the events since Sunday, in particular, have overridden that. They snowballed.
“But I have assured Chris that what we had verbally agreed I will honour.
“We’ll do the right thing by Chris as he has never done anything other than the right thing by me.”
Asked what had changed specifically since the demoralising result, Carter added: ““I think from previous defeats or poor performances there has still been an element of mitigation with what’s gone on.
“But I didn’t sense any feeling of that after Sunday.
“Something just didn’t feel right and Chezzy has admitted that to me as well.
“There’s something not quite right within the whole structure at this minute in time and something had to change.
“Ultimately, in this profession you can change three things: you can change the players, but that’s a difficult process given they are under contract.
“You can change the board and I’ve yet to find a madman that wants to fill my shoes so that’s a difficult one.
“Or you change the coach. As with everything, it’s a gamble.
“Who’s to say anyone will come in and do any better?
“We don’t know that, but what we are hoping for is some sort of spark or lift you do see when a change takes place and hopefully the players want to put a few wrongs right.
“We are always in their hands. By and large this season they have given everything they’ve got, but we just need some steel or toughness or something that just gets us over the line one week and I think the confidence would flow from that.”
Ex-Hull KR chief Chester helped hometown Wakefield to their highest Super League finishes in 2017 and 2018 but they have won only four games this year.
Carter - who said he had someone call about the head coaching role “within a minute” of the club statement being made public - feels the 42year-old understood the reasoning for parenting ways.
“He knows it’s a result driven business and ultimately whatever the reasons behind some of the results we’ve still only won four games out of 17 this season,” said the chief executive,.
“He understands that that's sport and he’s a head coach and he has to live and die by the results that come through.
“In the fullness of time, it might come as something of a relief as I know some of the things that he has had to put up with over the last two and half years and they are not pleasant.”
Poching, the former Leeds Rhinos and Hull KR assistant who was a legendary player at Belle Vue, now has chance to make his mark starting with Sunday’s game against another of his former clubs Warrington Wolves.
“We’ve got great confidence in him and we need to see a reaction from the playing group as ultimately they are the only ones that can change the situation we find ourselves in here,” added Carter.
“There’s all this talk about whether there will be relegation, won’t there be relegation but the facts of the matter are that, as things stand, relegation is very much the under-riding factor of this season.
“It’s not changed. And we only have four wins out of 17 and we need to start winning a few games.”
On Poching’s chances of staking a claim for the job, Carter - who is on holiday in Portugal - added: “It’s in his hands.
“I’ve only spoken to him briefly today - I get back into the country tomorrow morning - the way I see it now with seven games left in the season and if we were to win all seven then you’d kind of say that Willie’s earned the role.
“He’ll play his CV out for real over the course of the next few weeks.
“At the same time, whether we get a chance to meet some new candidates, someone who might impress myself and the board, over the next couple of weeks, we have nothing to rush with.
“Hopefully we can secure a few more wins that keep us safe. And then we have a break until the 2022 season and fundamentally - as we’re all aware with talks going on with SL and the RFL - 2022 is going to be a big, big season so we need to get someone in place who is going to be right. “That might be Willie or it might be somebody else.”