I agree with Lindy, M Ross, and Ros. I believe we should have the latitude to write out the full name of the month or to use the accepted three-letter abbreviation. I think genealogy is driven so much by the [numeric date] [3-letter month] [year] format that it's become ingrained in many of us. I know that when I'm making my own notes in any subject or context, that's what I use (except the naming of computer files; those always get yyyy-mm-dd numeric treatment so that they sort correctly).
But since the OP mentioned style guides, and since I have a fair number of them at my fingertips, I thought I'd include references. Among these seven, of note is that only The Associated Press Stylebook advocates that an abbreviation be used as the primary option, and then it always includes the abbreviation-delimiting period.
Ordered by title of publication:
Paula Froke, Anna Jo Bratton, Jeff McMillan, et al., editors, The Associated Press Stylebook, 55th Edition, 2020-2022 (New York, New York: Basic Books, 2020) 251.
"When a month is used with a specific date, abbreviate only Jan., Feb., Aug., Sept., Oct., Nov. and Dec. Spell out when using alone, or with a year alone."
The University of Chicago, The Chicago Manual of Style, 17th Edition (Chicago and London: The University of Chicago Press, 2017) 477, section 9.35.
"In text, therefore, the full date should always be spelled out (see 9.31). In documentation and in tables, if numerous dates occur, months may be abbreviated, and the day-month-year form, requiring no punctuation, may be neater (e.g., 5 Oct 2003)."
William A. Sabin, The Gregg Reference Manual, 11th Edition (New York, New York: McGraw-Hill, 2011) XL, "Essays on the Nature of Style," "Re: Abbrevs."
"Abbreviations of days of the week, of names of months, of geographic names, and of units of measure are appropriate only in business forms, in correspondence that is clearly expedient, and in tables where space is tight."
Kate L. Turabian (revised by Wayne C. Booth, et al.), A Manual for Writers of Research Papers, Theses, and Dissertations, 8th Edition (Chicago and London: The University of Chicago Press, 2013) 206, section 24.4.2, "Days and Months"
"In text, spell out and capitalize the names of days of the week and months of the year; see also 23.3.1. In tables, figures, and citations, you may abbreviate them if you do so consistently. (Note that some months in this system are not abbreviated.)"
Mark A. Stevens, editor, Merriam-Webster's Manual for Writers & Editors, 2nd Edition (Springfield, Massachusetts: Merriam-Webster Inc., 1998) 85, "Specific Style Conventions," "Dates."
"The names of days and months are spelled out in running text.... The names of months usually are not abbreviated in datelines of business letters, but they are often abbreviated in government and military correspondence."
The Modern Language Association of America, MLA Handbook, 8th Edition (New York, New York: Modern Language Association of America, 2016) 78, section 1.5, "Dates and Times"
"In the body of your writing, do not abbreviate dates, and be consistent in your use of either the day-monthyear style (12 January 2014) or the month-day-year style (January 12, 2014). In the latter style, the comma before the year has to be balanced by one after if there is no other punctuation after the year."
Anne Waddingham, New Hart’s Rules: The Oxford Style Guide, 2nd Edition (Oxford, UK: Oxford University Press, 2014) 179, "Abbreviations and Symbols"
"Names of days and months should generally be shown in full, but where necessary, as in notes and to save space, they are abbreviated thus: Jan., Feb., Mar...."
(Edited: Because Eric caught me in an error about the way I name my own computer files.)