Unfortunately, there is no indication as to why they make this claim, and as of this posting, inquiries to the city have gone ignored. It is probable that the proximity to the Trinity River is part of their claim. The Handbook of Texas Online has a little bit of background: (
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Rosser is just off State Highway 34 twelve miles southwest of Kaufman in southwestern Kaufman County. The area was settled before 1850 and was platted as a townsite called Trinidad by John S. Damron in 1851. The community was on the East Fork of the Trinity River, on a bend which provided a natural turning basin for the riverboats and barges that plied the river. As traffic on the river increased before the Civil War, Trinidad grew. A post office opened in the community in June 1854. Following the Civil War railroad construction increased in Texas, and river navigation declined. The Trinidad post office was discontinued in 1866. Construction of the Texas and Pacific Railway through the area of Trinidad, however, began in 1872 and immediately attracted numerous settlers to work on construction crews or to purchase land near the rail line. A boomtown of tents developed, and settlers rushed to purchase land in hopes of selling sand, water, or trees (used for crossties) to the railroad. The population once again justified a post office. In 1886 Capt. Robert S. Rosser, a local landowner and farmer who had profited through the sale of land to newcomers, the sale of timber to the railroad, and the operation of a commissary at the tent community, applied for a post office under the name of Burton. The office was granted, but, as there was already a Burton, Texas, the name assigned was Rosser.
Rosser grew rapidly, due largely to the presence of the railroad. In 1894 the town had a church, a cotton gin, a sawmill, and a gristmill, the latter three owned by Daniel M. McGee, the postmaster. A school with an enrollment of sixty-nine students was in operation by 1899. The community continued to grow during the late 1890s and into the early twentieth century. A second cotton gin was built, and a hotel opened for business near the railroad depot. Rosser supported, among other businesses, an insurance agency, a saloon, a general store, and, beginning in 1902, the Rosser Review. This newspaper remained in operation until sometime near the beginning of World War I, when the Rosser Midget began publication. The black residents of the community, most of whom worked for white landowners as farm laborers, were segregated into an area of Rosser called Red Row, after the red rental houses in the area. Rosser had a population of 128 in 1904 and 350 in 1925.
Oddly, Rosser had a small boom during the years of the Great Depression. Various new business establishments opened, three newspapers circulated, and electric power became available in the town. At least some of this growth was associated with a local businessman, R. W. McFarland, who sought to establish a pork-processing business in Rosser and went so far as to construct a granary in which to store corn for the hogs that he envisioned being shipped to the community from Iowa for slaughter and processing. Though McFarland's vision never became a reality, in 1936 the community had 350 residents and eleven businesses. Additionally, Rosser was incorporated during the decade.
Between 1940 and the mid-1960s the population of Rosser declined from 350 to 225, and its businesses decreased to five. At least a part of this decline resulted from the movement of residents to urban areas in search of jobs during World War II. The population of Rosser stabilized in the mid-1960s; it was 225 in 1976 and 255 in the late 1980s, when the community had one business. In 1990 the population was 366, and in 2000 it was 379.