Sure, but the Statute wasn’t drafted by the British government. It was drafted by the Imperial Conference of 1930, where the UK, the six Dominions, and India were all represented (“India” here meaning British India, administered by the British, of course).
The 1930 Conference built on the proposals from the 1929 Conference on Imperial Shipping laws (yes, major constitutional developments can come out of shipping conferences, in Brit-world).
The Conference was composed of the various PMs and other government ministers from each government. The terms of the Statute were referred to a committee chaired by Viscount Sankey, the Lord Chancellor, with representation from all the Dominions: the Inter-Imperial Relations Committee. The actual drafting of the statute in turn was referred to a sub-committee on drafting, chaired by Sir Robert Garran, Solicitor-General of Australia.
The name of the statute, based on the report of the Committee, was just mentioned in passing:
A special question arose in respect to the application to Canada of the sections of the Statute proposed to be passed by the Parliament at Westminster (which it was thought might conveniently be called the Statute of Westminster), relating to the Colonial Laws Validity Act and other matters.
That’s it for the name of the Statute: agreed upon unanimously by the Sankey Committee on Inter-Imperial Relations, with representation from all the Dominions, and then adopted by the Imperial Conference in plenary session.
No doubt the British parliamentary drafters put the bill into the final form for enactment by the British Parliament, but the decision as to the name was made by the Imperial Conference.
Summary of Proceedings of the Imperial Conference, 1930, printed by the order of the Parliament of Canada, p. 17.