Bacter is a Neo-Latin (i.e. Modern Latin) term coined from bacterium, which in turn derives from the Greek βακτήριον, meaning small staff (diminutive of βακτηρία).
[1] Nevertheless, for historical reasons, two archaeal species finish in -bacter: Methanobrevibacter and Methanothermobacter.
The Opinion opts for the latter: consequently, higher taxa are formed with the stem =bacter- and not =bactr-.
These rules were established so that the specific epithets were paired with the correct gender as imposed by the Bacteriological Code and the correct higher taxon names were formed.
[3] An interesting effect of this is that the genus Fibrobacter gives its name both to the phylum Fibrobacteres, which obeys Latin grammar, and to the class Fibrobacteria, which follows the recommendation of using the suffix -ia[4]