Can the original form of a familiar be detected by Truesight?

Further to the question "How to determine if an animal is a familiar or a regular beast?", can the original form of a familiar be detected by Truesight?

PHB pg 185 includes the following in the description of Truesight:

"perceives the original form of a shapechanger or a creature that is transformed by magic."

PHB pg 240 includes the following in the description of the Find Familiar spell:

Appearing in an unoccupied space within range, the familiar has the statistics of the chosen form, though it is a celestial, fey, or fiend (your choice) instead of a beast."

Where it gets more complicated is that also in the description of the Find Familiar spell, it indicates:

You gain the services of a familiar, a spirit that takes an animal form you choose

I found this question "Is a soul or spirit a creature?" regarding creatures and spirits, but I don’t believe the answer addresses whether a spirit is a creature.

I also found this question "What exactly is a fey fiend celestial spirit?" regarding spirits, but again I don’t believe it addresses whether a spirit is a creature (although it is very long so I may have missed something).

However, MM pg 279 includes this in the description of the Specter (Wights and Wraiths also have references to being spirits):

A specter is the angry, unfettered spirit of a humanoid

Accordingly, has the celestial, fey or fiend been "transformed by magic", and is its spirit a "creature", such that its original form would be apparent to someone with Truesight?