Poaching is less about the shell and more about the temperature.
Most boiled eggs- especially hard boiled- use boiling water- that is water at 212F or 100C at sea level. This is a vigorous boil.
Poaching, on the other hand, should stay at a low simmer. Somewhere around 160-180F. This is high enough to kill pathogens and denature proteins but because it is a lower temp, proteins cook more gently, will be more tender, and squeeze out less liquid.
Unfortunately, language is not so concise.
To make things confusing, soft-boiled egg recipes call for lower temperatures. Sometimes they are brought to a boil and then removed from the heat to allow to cook more gently. It would be accurate to say that this egg was poached. (Even though it would probably be referred to as a "boiled" egg)
As you have noticed, there can be overlap in the meanings of these words. You could say that you poached an egg in its shell if you were careful with your temps and wanted a tender egg (and you wanted to sound fancy). You would say you boiled your shell-less egg when you drizzle it into egg drop soup.
Egg that leaks out of the shell of a boiled egg is not called either because it is not part of the intended preparation.