The naming, since both versions are the IRS's own. Publication 969 says "FSAs are generally 'use-it-or-lose-it' plans. This means that amounts in the account at the end of the plan year can't generally be carried over to the next year." Notice 2005-42 says the same: "This rule is commonly referred to as the 'use-it-or-lose-it' rule, requiring that unused contributions or benefits remaining at the end of the plan year be 'forfeited.'" Notice 2013-71 uses the shorter form, in its own title and throughout. Neither spelling is a statutory term. Both notices flag it as the common name for a proposition whose actual legal source is the deferred-compensation ban, which is why no search of the Code for either phrase finds anything.
Run-out period versus grace period, which is where the money is actually lost. These two are routinely treated as the same thing and they do opposite work. Notice 2013-71 defines them side by side. A run-out period "is a period immediately following the end of a plan year during which a participant can submit a claim for reimbursement of expenses incurred for qualified benefits during the plan year." It then continues: "By contrast, a grace period is a period of up to two months and 15 days immediately following the end of a plan year during which a participant may use amounts remaining from the previous plan year ... to pay expenses incurred for certain qualified benefits during that two-month-and-15-day period."
So a run-out period is a paperwork window: it extends how long you have to file a claim, and the expense must already have happened inside the plan year. A grace period is a spending window: it extends the period in which you can incur new eligible expenses. An employee with unspent money on December 31 and a run-out period through March has no way to use it, because nothing they buy in January qualifies. The same employee with a grace period does. The two are also different in kind rather than in degree. A run-out period is a claims-processing window that a plan year assumes: Notice 2013-71 measures the unused balance itself "at the end of the plan's run-out period for the plan year." A grace period is one of the two reliefs an employer may elect, and a plan may offer neither.
Where the calendar arithmetic comes from. Notice 2005-42 created the grace period and set its outer edge: it "must not extend beyond the fifteenth day of the third calendar month after the end of the immediately preceding plan year to which it relates (i.e., 'the 2 and 1/2 month rule')." The notice then states the consequence in the form employees actually need: "The effect of the grace period is that the participant may have as long as 14 months and 15 days (the 12 months in the current cafeteria plan year plus the grace period) to use the benefits or contributions for a plan year." For a calendar-year plan that is March 15.
Leaving the job forfeits the balance, and there is one exit. Notice 2013-71 states that "any unused amount remaining in an employee's health FSA as of termination of employment also is forfeited (unless, if applicable, the employee elects COBRA continuation coverage with respect to the health FSA)." A health flexible spending account is a group health plan for those purposes, so continuation coverage can be available, paid for with after-tax money, and it is the only route by which a departing employee reaches a balance they have not yet claimed against.
Nothing converts to cash, ever. Notice 2013-71 is explicit: "A sec. 125 cafeteria plan is not permitted to allow unused amounts relating to a health FSA to be cashed out or converted to any other taxable or nontaxable benefit." Publication 969 says the same about the grace period, that the employer "isn't permitted to refund any part of the balance to you." An employer offering to pay out a leftover balance would be operating the plan outside section 125, and the consequence would fall on every participant's exclusion rather than on the one who was paid.
A carryover does not eat next year's election. Where a plan offers the carryover relief, Notice 2013-71 confirms that the amount carried forward "does not count against or otherwise affect the indexed ... salary reduction limit applicable to each plan year." So a participant can carry forward the permitted amount and still elect the full limit for the new year. The carryover cap is $680 and the salary reduction limit is $3,400, and the point here is that they do not interact.