I got myself a copy of the Clergy of Cashel, Emly and Leighlin (it was on sale and I'm aiming for a full set). These details don't exactly match.
Clergy of Connor (1993):
STEWART, RICHARD c1705-1778
b c1705 London: ed by Mr Clare London; ent TCD 8 Sep 1722 aged 17 Schol 1726 BA 1727 MA 1730 BD and DD 1750.
V Knockmark (Meath) 1729-47 and V Tara 1735-47; holding these with V Belfast (Conn) 1736-47; R Louth (Arm) 1747-78; Dean of Leighlin 1777-78.
s of Capt Robert Stewart of Coleraine.
Died Feb 1778 aged 73 buried in St Thomas Dublin.
Clergy of Cashel, Emly and Leighlin (2012):
STEWART, RICHARD 1704/05-1778
b 1704 or 1705; ed by Mr Clare in London; ent TCF 8 Sep 1722 aged 17 TCD Sch 1726 BA 1727 MA 1730 BD & DD 1750.
V Knockmark 1729-46; V Tara ((Meath) 1734-51; V Belfast (Conn) 1736-47; V Louth (Arm) 1747-78; Dean of Leighlin 1777-78.
d of Richard, gentleman.
Died Feb 1778 aged 73. Bur St Thomas's Ch Dublin.
Alumni Dublinensis:
STEWART, RICHARD, Pen (Mr Clare, London), Sept 8 1722, aged 17; s of Richard, Generosus; b. London. Sco. 1726. B.A. Vern 1727. M.A. Aest. 1730. B.D. and D. D. Aest. 1750. [Dean of Leighlin.]
So was Richard son of Richard or son of Robert? On the side of Richard is Alumni Dublinensis. On the side for Robert is Burke's Landed Gentry. Is there anything else on the Walsh family which refers to Mary Stewart's parentage?