I'm no specialist on poached eggs (don't eat 'em, but have made 'em for others) so the following are observations acquired at various times and places:
Eggs poached in water (vinegar optional, but can help them hold together) require water that is just simmering, no vigorous boiling thanks. Freshness of the egg is critical in getting a good consistency - the older the egg, the more ragged the effect. To assess freshness crack an egg onto a clean plate; if the yolk stands proud and the white stays together and slightly gelatinous = fresh, good for poaching. Yolk flattens out, white does similarly and is more watery = not so fresh, if poaching is necessary use a poacher/ramekin, but scrambling is a better proposition.
You can substitute a pea-sized bit of butter in your poacher for the oil if you prefer the flavour, but neither should really be needed.
Buttered eggs is a term that used to be used in the UK for scrambled eggs.
I suppose you could poach in butter at just the right temperature, but it would be very easy to get the butter too hot and end up with eggs deep-fried in butter, a cardiologist's nightmare, and not the poached egg you wanted.