For a standard four-egg folded homestyle omelette, the ideal size is a 10-inch skillet (measured by top rim diameter).
Your observations regarding the other sizes are entirely correct. An 8-inch skillet works wonderfully for two or three eggs, but four eggs create too thick of a layer, trapping raw liquid on top while the bottom overcooks and scorches. On the other end of the spectrum, a 12-inch skillet spreads four eggs out into a thin sheet that cooks too fast, dries out, and lacks the tender, custardy body needed for a satisfying folded omelette.
A quality 10-inch skillet typically provides an effective flat cooking floor of about 7.5 to 8 inches, surrounded by sloped or flared sides. This geometry allows four beaten eggs (roughly 200 to 240 milliliters of liquid) to settle at an optimal depth of roughly one-quarter inch across the base.
To ensure even cooking on medium heat, preheat your pan, melt a tablespoon of butter until gently foaming, and pour in the eggs. Use a heat-resistant spatula to push the cooked edges inward while tilting the skillet to let the raw mixture run onto the bare surface. Once the bottom is set and the top remains slightly glossy and soft, add any fillings and fold. The sloped sidewalls of the 10-inch pan will allow you to roll and slide the omelette onto your plate cleanly.