Common names for "rush" are usually related to a particular genus from a botanical family, for example:
Rush family – Juncaceae
Genus – Juncus
Species – Juncus effusus
Sedge family – Cyperaceae
Genus – Schoenoplectus
Species – Schoenoplectus lacustris
Bulrush family – Typhaceae
Genus – Typha
Species – Typha latifolia
Rush family – Juncaceae
Genus – Juncus
Species – Juncus effusus
Species – Juncus interior
Sedge family – Cyperaceae
Genus – Schoenoplectus
Species – Schoenoplectus acutus
Bulrush family – Typhaceae
Genus – Typha