Texas golf courses had a reputation, back in the days of Hogan and earlier, for dry, hard fairways. A player often had to land a ball short of a green to allow it bounce up onto the green. And when a player's ball stopped short of the green, the putter, rather than a wedge, might be a better choice for the next shot because of the firmness of the ground. So in Hogan-era Texas, the putter was often used from off the green rather than a wedge.
Texas' golf courses have come a long way since then, but the term stuck. A Texas wedge is the putter when used from off the green.
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