Sometimes arcane language is more precise than modern language.
The General Thanksgiving is found in the service of Morning Prayer. In Morning Prayer, Rite II, the prayer begs God to create in us an awareness of God’s mercies, that we might therefore be “truly” thankful. The same prayer in Rite I asks God to give us a “due sense” of God’s mercies, that our hearts might therefore be “unfeignedly” thankful.
Unfeignedly thankful is not quite the same as truly thankful. To feign something means to fake it, to put on the appearance of. Being feignedly thankful means to offer the sham of thankfulness. Which is what most people do on Thanksgiving. Feign thanks.