Rush said he used @localbod1 to correct the record from his perspective.
"But on other occasions, I was drawn into adopting the abrupt style and rhetoric of Twitter, which, albeit lacking profanity or malice, the comments fell short of my personal standards."
The account recently tweeted in relation to Wellington Water board appointments: "The other guy who was not approved is a former Labour staffer, left wcc when Justin lost, has great PR skills but no infrastructure experience. Well done Sean Rush."
The account's first tweet was an accusation that The Spinoff's Justin Giovannetti had unfairly paraphrased Rush's words at a council meeting, and it went on to post a series of tweets defending Rush and sharing controversial opinions on political issues.
Other tweets included posts that said climate change is "massively overstated", male politicians are likely to receive more political abuse, and suggested Māori co-governance is not part of the Treaty of Waitangi, according to The Spinoff.
The matter has weighed heavily on his conscience, Rush told Foster.
"It was poor judgment from me to maintain an anonymous account to address inaccuracies about me but I wanted to minimise the risk of personal attacks on me and my family, which as we all know is a prevalent part of politics today, especially on Twitter.
"I thoroughly regret this whole sorry saga and ask that you put it down to political naivety from which I will learn."
Foster accepted Rush's apology and stood by his previous comments to the Herald that said he thought it unwise for any elected official to run an anonymous account on any social media platform.
"It's plain to see that Cr Rush also came to this conclusion and while his actions left him open to criticism, he also deserves credit for publicly admitting the error of his ways. It is something all elected officials can learn from."
Councillor Rebecca Matthews has previously said the whole affair was an embarrassment for the council and it was deeply unethical.
"He is an elected person, he is accountable to the public for the things he says and does."