As per the AHS Hosta dictionary:

EYE – undeveloped or dormant shoot; a dormant bud. Frequently used for any division.

That means that as soon as you see anything above ground it is no longer dormant and in the process of developing and therefor no longer an eye, an eye you will only see when you dig up a hosta and examine the crown when the plant is still dormant. The AHS dictionary continues:

CATAPHYLLS – the often purplish short leaf-like coverings over the pips when they first emerge.

SHOOT – the stem and set of leaves of an individual division; the
dormant set of leaves all enclosed in a single set of leaf scales.

Pips doesn't make the dictionary as a defined term, although it does rate a mention under cataphylls.