“Principality” and “country” are not mutually exclusive; Monaco and Liechtenstein are principalities.
Regarding the union of the crowns, Poland was partitioned out of existence altogether in the past, and Croatia was subsumed into Yugoslavia; does that mean that Poland and Croatia are now not countries? Wales was conquered in 1282, but it wasn’t merged with England until 1536, so conquered =/= merged. Even that merger was clearly more show than substance: nearly 500 years later, Wales has its own language, history, culture, laws, legislative body (the Welsh Assembly), international representation (e.g. the Wales team in World Cup soccer or Six Nations rugby).
It’s not an independent nation-state, but I don’t think the 1282 conquest or Charles’s ceremonial title “Prince of Wales” are proof that Wales is not a country.