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Are nylon guitar strings sold in a pack such that only first 3 are nylon, rest steel?

Recently ordered a pack of nylon strings, and on opening the pack realized that only the first 3 strings are nylon, but the last three (4th, 5th, and 6th) are wire-wound, and appear to be steel core. However, they are quite whitish in-spite of the metallic appearance (most of the steel-core wire-wound strings are brassy look). Have I been cheated ? Is this a substandard set ?

The guitar in question "qualifies" to a large extent to be called a "classical guitar", because

  • Has higher than usual nut and bridge, with much more action than one expects on regular steel-string guitars
  • Has an absolutely flat fretboard, that is certainly wider than my steel-string guitar
  • Has a bridge to which the strings can be tied, although it does have a tail-bracket into which steel-strings with ball-ends can be fitted quickly.
  • Has tuning posts which look like this enter image description here

instead of tuning posts like this

enter image description here

on my steel-string guitar

However, this is a guitar I bought used, some 30 years ago, and the seller had steel strings. It was my first and only guitar for next 10 years, so I've to admit that I am not 100% sure if it is indeed a real classical guitar.

Also, after having strung the guitar with the new pack, I've realized that the strings are indeed nylon (even 4th, 5th, 6th, even though wirewound are not stiff and hard to tie into a knot, as the steel string... which is in fact impossible to tie into a knot without snapping them). However, it was a nightmare to get the guitar to tune, and it just refuses to stay in tune for 5 minutes. It wasn't the case while the guitar was strung with steel strings. This could mean that it might note be a classical guitar then, although I'd have though that a steel-string guitar could be strung with nylon string, but not vice-versa.