Weisstein states "The smallest known area shared by three primitive right triangles is 13123110, corresponding to the triples (4485, 5852, 7373), (1380, 19019, 19069), and (3059, 8580, 9109) (Beiler 1966, p. 127; Gardner 1984, p. 160), as discovered by C. L. Shedd in 1945." He doesn't define primitive right triangle in the online material.
Eric W. Weisstein, World of Mathematics, 1999, CRC Press LLC.
In the following material the matrices that transform one Pythagorean triple into another appear. The logical inference is that a primitive right triangle is one which cannot be obtained by such a transformation. One such is that associated with the triple 3, 4, 5.
References (by Eric W. Weisstein)

Beiler, A., "The Eternal Triangle." Ch. 14 in Recreations in the Theory of Numbers: The Queen of Mathematics Entertains. New York: Dover, 1966.

Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 121, 1987.

Dunham, W. Journey through Genius: The Great Theorems of Mathematics. New York: Wiley, pp. 120-121, 1990.

Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 160-161, 1984.

Guy, R. K. "Triangles with Integer Sides, Medians, and Area." D21 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 188-190, 1994.

Kern, W. F. and Bland, J. R. Solid Mensuration with Proofs, 2nd ed. New York: Wiley, p. 2, 1948.

Ogilvy, C. S. and Anderson, J. T., Excursions in Number Theory. New York: Dover, p. 68, 1988.

Rothman, T. "Japanese Temple Geometry." Sci. Amer. 278, 85-91, May 1998.

Sierpinski, W. Pythagorean Triangles. New York: Academic Press, 1962.

Whitlock, W. P. Jr. "Rational Right Triangles with Equal Areas." Scripta Math. 9, 155-161, 1943a.

Whitlock, W. P. Jr. "Rational Right Triangles with Equal Areas." Scripta Math. 9, 265-268, 1943b.
The following shows the transformations by matrices of Pythagorean triples into other Pythagorean triples, something expressed in the online pages I was led to by Google search on "primitive right triangles." This, part of a Mathematica session, shows the transforming matrices in list form, row by row. 
[Graphics:Images/primitive_tria_12_19_02_gr_1.gif]
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Converted by Mathematica      December 19, 2002

12/27/02 Version http://www.cs.ucla.edu/~klinger/primitive_tria_12_19_02.html
©2002 Allen Klinger