BRITAIN =   the ISLAND of BRITAIN, not any of the islands around it, ie this excludes certainly the IOM, but also the Isle of Wight, Lindisfarne and all the rest. These are covered by the term GREAT BRITAIN.

Northern Ireland is included in the term GB & NI or UK.

There has been no proper use of the term BRITISH ISLES since Irish Independence, 99 years ago- before that that ALL the archipelago belonged to the London government, ie it was British.
The adjective British has been retained in independent countries, like in British Columbia, but that's their choice and I don't the Irish would go for that?

Any use of the BI term, even geographically, is misleading and incorrect and it should not be used.