The Scottish church that focused Edinburgh’s informal Hogmanay street party before 1993 was Tron Kirk.
The former parish church stands on Edinburgh’s High Street in the Old Town, close to the Royal Mile. Its prominent clock and spire helped make it a natural meeting point: the bells could be heard and the building could be identified across the crowded streets. The gathering was a popular, largely informal celebration rather than the ticketed city event familiar today.
Tron Kirk was built in the 17th century and formally opened in 1641. Its name comes from a Scots word for a public weighing beam, which once stood outside the church. Although the church closed in 1952, the site retained its association with New Year celebrations.
A common mix-up is St Giles’ Cathedral, Edinburgh’s best-known historic church. St Giles’ is nearby and highly significant, but the traditional Hogmanay gathering point described in this question was specifically the Tron Kirk. In 1993, organised festivities shifted toward Princes Street.