Based on the available materials, the following is a brief summary of the information provided:
Summary of Available Information: The sources exclusively define the term "engagement." Key definitions include: * A mutual promise to marry: The most relevant definition in the context of the search query, describing a betrothal or promise of marriage. * A period during which this promise is kept: Referred to as a "long engagement." * An appointment or arrangement: A commitment to be at a particular place or time, such as a "dinner engagement." * Employment for a specified time: Such as a "speaking engagement" for a performer. * A hostile encounter; a battle: A less common definition related to military conflict. * The act of engaging or the state of being engaged: This includes the mechanical sense (e.g., the engagement of a clutch or gears) and the social sense (e.g., engagement in diplomacy or social work).
The sources also note the existence of an "engagement ring" as a token of betrothal but provide no details about its design, history, or gemstones.
Conclusion: No information about gemstones, birthstones, or specific jewelry designs was found in the provided sources. Therefore, a detailed article on the requested topic cannot be written. The available data is limited to linguistic definitions of the word "engagement."