Unseen Targets and Attackers:
When you Attack a target that you can’t see, you have disadvantage on the Attack roll. This is true whether you’re guessing the target’s location or you’re targeting a creature you can hear but not see...When a creature can’t see you, you have advantage on Attack rolls against it. (PHB p.194)
Thus, the advantage/disadvantage mentioned for invisible creatures comes from being unseen, which is negated by True Sight. Since the character with True Sight can see the invisible creature, it no longer has the "invisible condition" in respect to the character with True Sight, but for everyone else it still does,
The example spell you gave (Faerie Fire) is a different case. Faerie Fire cancels the benefits of being invisible, but does not cancel the invisibility.IE. You see the faerie fire outlining the creature, but do not see the creature. And since everyone can see the faerie fire, everyone knows where it is.
Summary:
The two spells are worded differently, because Faerie Fire cancels the benifits of being invisible for everyone that can see the fearie fire, while True Sight only cancels the invisible condition for the character that has True Sight.
Yes, the sensor can be seen with truesight
I will briefly state what has already been pointed out in Dinomaster's answer:
The clairvoyance spell specifies that it creates "an invisible sensor", and truesight it described as allowing a creature to "see invisible creatures and objects", so this implies that such a sensor can be seen via truesight, although since a sensor isn't explicitly described as an object, it's not conclusive evidence...
In addition to that, a similar situation exists for the spell scrying (PHB, pg. 273), which is made more explicit:
... the spell creates an invisible sensor within 10 feet of the target. [...] A creature that can see invisible objects sees the sensor as a luminous object about the size of your fist.
So not only can creatures with truesight (or any creature that can see invisible objects for any other reason) see the sensor, but according to scrying, it looks like "a luminous objects about the size of your fist".
Given that the language regarding the "invisible sensor" is the same in both spells, it seems as though invisible sensors can be seen by creatures with truesight, and furthermore it's likely that the appearance of the sensor described by scrying would be the same for clairvoyance as well (or at the very least it gives the DM a precedent to follow).
Best Answer
Yes
Truesight specifies (PHB, 284). Emphasis is mine.
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.