An agreement to marry is an engagement.
While a ring may signify an engagement, it is not the thing that makes the agreement.
An engagement ring is, however, a conditional *gift*, which must be returned if the engagement is broken. It is the property of the receiver as long as the condition of the gift is fulfilled, but the giver if it is not. That is a legal concept that will hold up in court, and has.
Per etiquette, being "engaged" really means you have made an agreement, and are actively planning the wedding and reception, which will beheld at the earliest possible. It doesn't mean you're playing house or pretending to be married to try it out for a few ,more years.
But yes, the agreement on both sides to marry is the engagement, and a ring is just a nicety. You ARE engaged even before you've bought a ring- which you would hopefully do together.