Electrical – How to one make a connection where flexibility is necessary despite the connection being exposed to physical damage

It's up to the AHJ to make the call. The term is used to express that some methods and materials offer better protection against physical damage, and that those methods and materials should be preferred when there's potential for greater than normal physical damage.

I'd suspect you wouldn't encounter severe physical damage in residential settings often, though it may be more common in industrial settings where there is equipment and machinery moving hither and thither.

After further research, it appears the term was approved at some unknown time. It was possibly written in as part of the original proposal for an article, and then propagated as similar articles were proposed.

There have been proposals to apply the term to other areas of the code. Though as far as I can tell, all such proposals have been rejected. The outcome of one such proposal can be found in this document, where the submitter proposes a revision to 330.12(1) to include the term.

The uses not permitted section of Article 330 should acknowledge the protection against physical damage provided by the metallic covering of this wiring method by limiting the cable from use where it is subject to severe physical damage (such as forklift or vehicular damage). The degree or magnitude of physical damage depends on the wiring type and the likelihood that damage prior to or after installation could severely damage or cause the wire to be inoperable. Type MC Cables have a metallic covering which provides protection to the inner insulated and non-insulated conductors against physical damage before, during, and after installation.

While the term “severe physical damage” in not defined in the NEC and thus requires evaluation by the AHJ, the phrase is used to qualify the suitability of several wiring methods including in 358.12 for EMT, in 368.12 for busways, in 370.7 for cablebus fittings, in 376.12 for metal wireways, in 380.12 for multioutlet assembly, in 386.12 for surface metal raceway, in 388.12 for surface nonmetallic raceway and in 392.12 for cable trays.

A comment from one of the folks who voted to reject the proposal, basically points out that the term only currently applies to raceways and wireways not cables.

... The submitter’s comparison to articles 358, 368,370, 376, 380, 386, 388, and 392 all include wiring methods or systems that are unique from that of cable either by virtue of limited use or they are metal raceways or wireway. Cables introduce a significant degree of flexibility in the manner and location that they are routed and the proliferation of their use within the industry.

Everybody involved agrees that the term is undefined, subjective, and the final determination as to if it's "severe" physical damage is left up to the AHJ.

If an area rodents can infiltrate is considered subject to physical damage, then cables run inside finished walls, between joists in attics, etc. would certainly have to be considered to be subject to physical damage.

Everything is subject to physical damage in the strictest sense. In context, it's clear the code means subject to physical damage in the normal usage.