Sebastian Adrian Nowak with an address at Colvil Street, Belfast, is accused of murdering Mr Coulter (62) from Sandy Braes between October 13 and 18.
Nowak also faces a charge of assaulting a police officer on October 18.
A detective sergeant told Londonderry Magistrates Court today that he believed he could connect the Nowak to the charge.
Nowak spoke only to confirm his identity and that he understood the charges.
There was no application for bail and while details surrounding the incident were not disclosed it emerged Nowak was in breach of bail terms on unrelated matters by being in Magherafelt.
A defending solicitor explained a lot of rumours are circulating on social media which are "patently untrue” and it was important this was placed on the record.
He said: “My client instructs he acted in self-defence and wishes to extend his condolences to the family and friends of the deceased.”
Deputy District Judge John Connolly remanded Nowak in custody to appear before Magherafelt Magistrates Court by video-link on November 17.
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