As for a google search: "Cayman trough" gets 9920 hits whereas "Cayman trench" gets 14,800 (note the " " were in the search to get a direct match). That is a bit more, but look at the quality of the hits: trough gets more scientific hits versus trench getting more general hits. Interesting to note that the Wikipedia article is the first link ...

Tectonic setting. The eastern part of the Cayman Trough forms part of the Gonâve Microplate, which lies between the North American Plate and the Caribbean Plate.It is bounded to both north and south by large transform faults that together accommodate the relative displacement of the two major plates. To the north the boundary is the western part of the Septentrional-Oriente …

La Cayman Trough (noto anche come Cayman Trench, Bartlett profonda e Bartlett Trogolo) è un complesso trasformare guasto zona bacino di pull-apart che contiene una piccola cresta diffusione, l' aumento Mid-Cayman, sul pavimento del western Caraibi tra Giamaica e le Isole Cayman. È il punto più profondo del Mar dei Caraibi e fa parte del confine tettonico tra la …

Ten Brink, Uri S., et al., 2001, Asymmetric seafloor spreading, crustal thickness variations and transitional crust in Cayman trough from gravity, GSA Meeting Abstract; Roberts, H. H. 1994 Reefs and lagoons of Grand Cayman Monographiae biologicae (Brunt, MA; Davies, JE eds). Kluwer Academic Publishers, Boston, Ma The Hague. ISBN 0-7923-2462-5

The Beebe Hydrothermal Vent Field (also known as the Piccard Vent Field) is located just south of Grand Cayman in the Caribbean, on the north side of the Mid-Cayman Spreading Centre in the Cayman Trough. Approximately 24 kilometres (15 mi) south of Beebe is the Von Damm Vent Field.. At nearly 5,000 metres (16,000 ft) below sea level, Beebe is the world's deepest known …