Truesight specifies (PHB, 284). Emphasis is mine.
A creature with truesight can...automatically detect visual illusions and succeed on saving throws against them, and perceives the original form of a shapechanger or a creature that is transformed by magic. Furthermore, the creature can see into the Ethereal plane.
Invisibility is an Illusion spell(PHB, 254), and Truesight automatically detects visual illusions and gives you perception of a creature trasnformed by magic. Therefore, you not only see them, but you've detected the illusion of invisibility or seen the original form of a creature that uses magic of any sort to make them invisible.
For See Invisibility, The rest of the party may also have input
The language remains a bit of an issue for See Invisibility, but as long as there is one more person available to ask 'Hey, you see this?", then they'll confirm the creature is invisible.