Interesting that you specify Lippia graveolens as Mexican oregano. That species is actually in family Verbenaceae and is native to Texas and Mexico. That make it a close relative of thyme. The other spices being discussed are all Mint family, now called Lamiaceae.

I find 5 species in Origanum listed, these include the classic oregano and marjoram. However, Greek oregano is probably a sage, genus Salvia.

There are 95 species of Salvia listed, Salvia officinalis being the classic culinary sage or "kitchen sage".

There is no way to get through the confusion when things are marketed under common names and not scientific names. Mexican sage and Mexican oregano are used interchangeably and can refer to Lippia, Origanum, and probably as many as 5 species of Salvia. Seems you need to buy the type you like from the same source each time.