Your friend is oversimplifying things tremendously.
Classical Guitars vs. Steel-string Acoustics
To begin with, a modern classical guitar is an acoustic guitar strung with nylon strings (in the past, they were strung with strings made from animal gut). It differs from a modern "flat-top" acoustic guitar in that flat-top's are strung with steel strings, which have a much higher tension than nylon strings. For this reason, the internal bracing of the guitars are much different as well, and their tones are quite distinct.
Playing a Guitar in the Classical Style vs Fingerstyle
Both "classical" style and fingerstyle guitar playing involves plucking the strings with the fingers rather than with a pick. However, the classical style has, over time, developed a widely-accepted orthodoxy to its technique. This involves using the thumb and the first three fingers of the right hand, but not the pinky. All four fingers can be used to pluck any of the strings.
Fingerstyle playing, by contrast, is the result of folk music traditions, and has fewer commonly accepted "rules". There are sub-genres of fingerstyle, such as Travis picking, in which the thumb plays the bass notes and the fingers play the melody strings, but even there there is no general agreement on how many fingers to use. I use the first three fingers, for example, while Doc Watson uses only his index finger. Modern fingerstylists will do whatever they can to extract sound out of the instrument, including tapping the fingerboard, striking the strings or the body for percussive sounds, etc.
Generally speaking, you can play either style on either kind of instrument. Chet Atkins, for example, played fingerstyle but often using a classical guitar. Some classical techniques don't work as well on steel string instruments, and vice versa, but for the most part you can use either style with either instrument.
I don't think this is what you're really asking about, but here's a brief definition: Classical music is music from or based on the European art music tradition. This stretches back many centuries, to Gregorian chants and before, and has evolved through many different significant "periods" (such as the Baroque, Classical, Romantic, etc.) to the present day.
Classical music is not to be confused with the Classical Period of music history. The Classical Period refers to the style best exemplified by Haydn, Mozart, and early Beethoven, and comprises at most about a fifty-year period of music history. It is just one of many periods in classical music.
Classical music is distinct from, say, Jazz or Blues in that, for the most part, it lacks much, if any, African influence on its sound. This has changed somewhat in the last century or so, but such exceptions represent a small portion of the overall classical tradition.