According to the SupernaturalWiki, these are the uses of the Colt:
- 6.18 Frontierland - Samuel Colt shoots and kills two demons (in 1861).
- 6.18 Frontierland - Dean shoots and kills the phoenix.
- 1.20 Dead Man's Blood - The Colt is introduced with 5 (out of 13) remaining bullets. John shoots Luther the vampire to save Sam's life.
- 1.21 Salvation - Sam shoots at the Yellow-Eyed Demon in Rose's nursery.
- 1.22 Devil's Trap - Dean shoots the man possessed by a demon (Tom) to save Sam's life.
- 1.22 Devil's Trap - Sam shoots John in the thigh while he is possessed by the Yellow-Eyed Demon.
- 2.22 All Hell Breaks Loose: Part Two - Dean kills Azazel with a shot in the heart.
Also, according to John Winchester,
Story goes, he made thirteen bullets. This hunter used the gun a half dozen times before he disappeared, the gun along with him...
So your count is right, 3 + 6 + 5 = 14.
But it's still possible that there were only 13 bullets. Depending on context, "a half dozen" can mean any quantity from 5 to 7. A bit like "several" and "a couple". So if in this case it actually means 5, your count would end up 3 + 5 + 5 = 13.
Of course, another but less likely possibility would be that Samuel Colt, being the sharpshooter that he is, shot both demons with one shot.† That way, it would be 2 + 6 + 5 = 13.
†: I've never actually seen the episode where that happens, so I don't know if this is a plausible or even possible explanation.