I think the confusion arises from the original 'horse guards' who later added horse grenadiers, the original horse guards were disbanded and the horse grenadier troops went on to become the life guards.
our horse guards (the Blues)were originally the 1st regiment of horse or the 'Earl of Oxford's horse' until 1746 when they lost their number, and a few years later became the Royal horse guards, and till 1820 were regarded 'unofficially' household cavalry.
The information you gave is on the regiments.org site.
for most of the 18th C. they were brigaded with other line regiments.
Richard Davenport (1719-1760) who was in the original 4th troop of horse guards, refers to them as part of 'the horse' as distinct from the household brigade, or just calls them the 'Blues'
hope that helps muddy things