This style is directly decended from the psalm singing that the earliest settlers in this country were doing. If done with a song leader quickly giving out the lines before the singing, it's generally called lining out. If the words are sung from the book or memory 'old way' is the most consistenly used label.
The texts are primarily the old psalms, Watts' psalms, Wesleyan hymns, & some folk hymns.
If Jean's church did not sing out of the following book, they probably used a similar one: The Sweet Songster, A Collection of the Most Popular and Approved Songs, Hymns, and Ballads by Edward W. Billups. "This is one of the hymn books used by the Old Regular Baptists. It was evidently first published by Billups in 1854. It contains 350 pages on which we find 285 numbered hymns, followed by five "choruses," then six more hymns that are not numbered, then "A Form of Matrimony" (a short marriage ceremony), and a first lines index." A soft cover reprint is available from Lillian M. Arrowood, P.O. Box 398, Wayne, WV 25570 and costs $4.00 per copy, plus postage of $3.00 for up to six books.