The demonym "Abyssinian" was traditionally only used for Habesha peoples. Although, it expanded to also include Oromos & some others as Abyssinia expanded.
Actually, the term 'Abyssinia' referred to the ruling Christian realm in Ethiopia wich included the Amxaras and Tigrayans + a few other Christian ethnic groups. It's a term made up by Europeans though, Ethiopians never called themselves 'Abyssianian' nor 'xabesha'. Even 'xabesha' is a word the Arabs first used to describe the Axumite invaders and changed menaing throughout the course of history.
But in general, both the terms 'xabesha' and 'abyssinia' were used interchangeably to describe the ruling Amxara, Orthodox Christian culture. After the decline of Axumite kingdom the Tigrayans lost control and got replaced with an Amxara dynasty, the Zagwe.
Now, this is a whole different topic but briefly, the Amxaras, who were southern Cushites/Agaws were serfs and low caste people in the Tigrayan society of Axum. After the Zagwe adopted Axumite culture they gave birth to the Amxaras and continued to rule for many years. The Tigrayans did get back some control later on but it was an on and off thing thorughout history. Amxaras/Agaws has always been separate from Tigrayans.
EDIT: Amxara culture in its basis is a culture of expansion and assimilation. The Amxara royal family accpeted and included any Cushitic group that was Orthodox Christian. In fact many Oromo rulers ruled Ethiopia, such as Selassie and Menelik. However, Oromo culture wasn't valued. Once you're Amxara and Christian that's all u are. When Amxaras converted Gallas/Oromos they made them speak Amharic and dress like an Amxara. Everything Amxara.