Or are they instead called “applications” in Britain? One hears of motions filed in court cases, perhaps primarily in the U.S. But do these exist in Britain, or in any other jurisdictions? Or are they called something different?
The term "motion" is not used in court rules of procedure in England and Wales.
'application notice' means a document in which the applicant states their intention to seek a court order
The process itself (rather than the document) is called an "application." See e.g. Practice Direction 3A relating to the process for seeking a court order to strike (dismiss) a claim that has no reasonable grounds of success or for summary judgment. It uses the phrase "make an application for an order."
See also Thomson Reuters Practical Law Glossary.