Trump
admitted, in that NBC interview, that disliking the Russia investigation was at least part of his motivation to fire Comey.
That doesn't mean that firing him was actually sufficient to stop the investigation. To stop the FBI's investigation, Trump would have to nominate a strong ally as the next FBI director, and Congress would have to confirm them. Getting Congress to confirm a partisan nominee would be tricky to start with, and much harder with the media furor.
Separately from (but complementary to) the FBI investigation, there's the question of a special prosecutor, which can be appointed either by Congress (making them part of the legislative branch and independent from the executive branch), or by the Justice Department (which is nominally under the president's authority, but only in the sense that Trump could fire Rosenstein and cause an even bigger shitstorm). The latter is what just happened.
January 24th is when the FBI interviewed Flynn. On the same day, anonymous sources told the press that they had found "no evidence of impropriety" in the phone calls - but that's different from the FBI formally closing the investigation. Among other things, it leaked much later (around February 16) that Flynn had falsely told his interviewers that he hadn't discussed sanctions in those calls, whereas the recordings showed that he had. This potentially opened the door to charges of lying to law enforcement. The second round of leaks said that the FBI believed he was trying to be "truthful" and wasn't intentionally lying, so was unlikely to be charged, but again, that's different from formally closing an investigation. Even now, the broader investigation into Trump-Russia ties continues, and Flynn remains a part of that investigation.