Strictly speaking, a type species exists only in zoological nomenclature. As set in Art 42.3 of the ICZN, the type of the name of a genus or subgenus (a “genus-group name”) is the "type species". This is defined as "The nominal species that is the name-bearing type of a nominal genus or subgenus". The species name in turn is fixed to a type specimen.
In botanical nomenclature, the type of a name of a genus or species is a specimen (or illustration) (ICBN, Art 10.1, 8.1 and 10.4). This type will usually be the type of an included species, in which case it can be indicated by the name of this species (Art 10.1). This species is called the "type species", but this phrase has no formal standing. The botanical type specimen itself is generally called a "type specimen."
Thus, the two definitions are generally identical in applied use, but defined differently.