Did the british name New York and Rhode Island?

Yes, the British named both New York and Rhode Island.

New York was originally named New Amsterdam by the Dutch, but the British changed the name to New York in 1664 when they took over the colony.

Rhode Island was originally called Aquidneck Island by the Native Americans, but the British changed the name to Rhode Island in 1644 when they established the Colony of Rhode Island and Providence Plantations.