The Republican Party was never a ‘third party’. In 1854 (before the 1860 election) when they where formed, they where so popular, they immediately pushed the Whig party down to third party status. They have never held the position of ‘third party’ option.
Of course they were a third party. The fact that they gained prominence quickly does not change that.
Third party Def.
(in a two-party system) a political party formed as a dissenting or independent group from members of one or both of the two prevailing major parties.
From 1836 - 1850 the two party system was Democrats and Whigs.