The term "royal" in this context specifically means the "capture/mate me to win" property. The term "prince" has been used both for pieces with the King-move but no such property (Tempête sur l'Échiquier comes to mind) and for pieces that serve as a backup royal (Chu Shogi, Tamerlane). Prince is a reasonable term for either, but the clarification is also reasonable.
The term "royal" in this context specifically means the "capture/mate me to win" property. The term "prince" has been used both for pieces with the King-move but no such property (Tempête sur l'Échiquier comes to mind) and for pieces that serve as a backup royal (Chu Shogi, Tamerlane). Prince is a reasonable term for either, but the clarification is also reasonable.