I’m not an expert on the rigging of American ships, being from the UK, but the differences with English practice should not be that great. There are two lines on the pictures you show. The block in your first picture relates I think to the topsail yard tye.
Blocks are stropped to the Topsail yard and Cross trees thro' which the tyes run port and starboard.
The tye starts at the mast head above the cross trees, runs down between the centre of the crosstrees, reeves thro’ the yard tye blocks, up thro' the upper tye blocks and continues down on a long pendant with a double block seized in it end.
The Topsail Tye halliard starts at the channel to a block seized to an eyebolt.The line runs up thro' the pendant block then run down thro’ the channel block back up thro’the pendant block and down to the deck and belays on a kevel inside the bulwark.
This is the tackle shown in your second picture.
The Channel tackle shown in your third picture relates I think to a Topmast breast backstay which in the pictures appears to pass close by the topsail tye tackle hence possibly the confusion, but the two lines are not associated. The breast backstay is part of the Standing rigging, and the topsail tye part of the running rigging.
Hope this helps