The Chicago Manual of Style's treatment of abbreviated dates addresses two aspects of the subject: informality and ambiguity. The guideline in the sixteenth edition (2010) that you allude to in your question mentions both of these aspects, but it focuses primarily on ambiguity:
6.106 Slashes with dates. Slashes (or periods or hyphens) are used informally in all-numerical dates (e.g., 3/10/02), but this device should be avoided in formal publications to prevent ambiguity: Americans usually put the month first, but other countries do not (e.g., Canadians and Europeans put the day first). If an all-numeral format must be used, use the ISO standard date format (year, month, date, in the form YYYY-MM-DD; [cross reference omitted]).
Evidently, Chicago views the all-numerical slash style of date writing as being informal—which may or may not create a problem in tone, depending on how formal the surrounding text is—but the more serious problem from CMOS's perspective is that readers may not be able to tell whether (for example) 4/5/2017 refers to April 5, 2017, or to May 4, 2017.
A subsequent CMOS guideline suggests the ISO format (again) but also other unambiguous short forms:
9.36 All-numerical dates and other brief forms. ... When quoting letters or other material dated, say, 5/10/03, a writer must first ascertain and then make it clear to readers whether May 10 or October 5 is meant (not to mention 1903 or 2003). In text, therefore, the full date should always be spelled out [cross reference omitted]. 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).
In this guideline, although the imperative to avoid ambiguity again receives attention, the governing idea seems to shift to formality versus informality. After all, there is nothing ambiguous about the date 5 Oct 2003—and yet CMOS endorses its use only "in documentation and in tables" while insisting that in main text the full date always be spelled out in full. The "therefore" that CMOS inserts before making its formality-based recommendation is, therefore, clearly specious. Nevertheless, it is CMOS's advice, so if you are following that style guide, you had best reconcile yourself to spelling out dates in main text and limiting your use of short forms of the type "5 Oct 2003" to notes, tables, and bibliographical entries.
Finally, CMOS identifies three alternative short forms for the months of the year, as follows:
Jan., Feb., Mar., Apr., May, June, July, Aug., Sept., Oct., Nov., Dec.
Jan, Feb, Mar, Apr, May, Jun, Jul, Aug, Sep, Oct, Nov, Dec
Ja, F, Mr, Ap, My, Je, Jl, Ag, S, O, N, D
CMOS offers the following assessment of these three systems:
10.40 Months. Where space restrictions require that the names of months be abbreviated, on of the [three preceding] systems is often used.The second and third, which take no periods, are used respectively in computer systems and indexes of periodical literature. In formal prose, Chicago prefers the first.
Once again, opposition to perceived informality (not ambiguity) leads Chicago to oppose using abbreviated months in main text and, indeed, anywhere else not excused from the normal rule owing to "space restrictions" (which in practice usually refers to tables, charts, infographic labels, footnotes, and bibliographies).
To return to your original question, CMOS offers the following style guidelines for handling dates:
1. In main text, always spell out the month and all four digits of the year: January 1, 2014.
2. If you must use an all-numerical date format, follow the ISO standard YYYY-MM-DD format: 2014-01-01.
3. If space restrictions (in tables, charts, notes, etc.) require you to abbreviate the month, you may use 1 Jan. 2014, 1 Jan 2014, or 1 Ja 2014—but Chicago prefers the first of these styles.
It is worth emphasizing that Chicago cites (at 10.40) the periodless "1 Jan 2014" format as the one favored "in computer systems"—so if you are using dates in the context of computers, you might follow CMOS's hint and adopt the "1 Jan 2014" format as more appropriate to your particular context than the "1 Jan. 2014" format would be.