I think that the 1710 record referred to is on Family Search & Scotland’s People but sadly no first name recorded)
? Armour birth: 16 September 1710
Baptism 19 September 1710, KILMARNOCK, AYR, SCOTLAND
Residence: 1710 Kilmarnock, Ayrshire, Scotland – Father James
You could check the record on SP it might give a clue to see if it was William.
William Armour married Margaret Wylie 7 June 1736, Kilmarnock
It seems they only had three children, Andrew 1737, Agnes 1739 – Margaret age 0 burial 22 Nov 1746 father William.
It’s likely that Margaret Wylie was baptised 12 Aug 1705 in Kilmarnock parents Thomas Wylie & Isabel Legate but there are other options elsewhere in Ayrshire.
(Also Thomas 1699, John 1702 & Jean 1712)
The only record on SP in Scotland for:
William Armour baptised 4 June 1702 in Kilmarnock parents David Armour and Elspeth Park. (Also David 1700 & Robert 1705)
The snag with both baptisms in the naming of their children
Colin