During Strength of Materials class, when I was at the other end of my career, we were taught that theoretically there was no such thing as a non-extensible material; such a material would be so brittle that it would snap as soon as a tensile force was applied to it - like a very short pastry.

But in general use, as you say, the distinction is far from clear. Extensible seems to me to indicate stretchability whereas extendable suggests an opening out or lengthening. An elastic band is extensible, ie: it can be stretched, whereas my lunch hour is extendable, ie: it can be added to, but an hour is an hour and cannot be stretched. That may be a purely engineering view of it though, we do value our lunch hour.

There is also extensile which seems to mean the same as extensible.