Greek Oregano~ Origanum vulgare hirtum
How to know if you have Greek oregano or common oregano is by taste and flower colour. The taste of a fresh leaf is well peppery and leaves an almost burning sensation on the tongue (this is good, this what you want!), whereas common oregano, although OK in its own right, is a lot blander.
The other difference between the two is the flower colour, common being pink and Greek being white.
The plant I have here ticks all the boxes, your search is over!