"The Old Kingdom" isn't an ancient term for the period, but rather a term coined by 19th century historians. Strangely enough (In my personal opinion) it doesn't even include the first 2 dynasties, but starts at the 3rd and lasts until the end of the 6th - which is where the date 2181 BCE comes from. 2181 is the last year of the reign of pharaoh Nitiqret, which we know from the Ramessid king list (the Turin Canon). After that the pharaonic rule over Upper and Lower Egypt disappears during the 1st intermediate period (dynasty 7-10, or 2160-2055 BCE).

According to Jaromir Malek in Ian Shaw's The Oxford History of Ancient Egypt, all pharaohs during "the Old Kingdom" were related to Queen Nimaathap who was the queen of pharaoh Khasekhemwy, the last pharaoh of dynasty 2. Thus the date 2686 is derived from the first year of her son, pharaoh Djoser, the first pharaoh of the 3rd dynasty. A more noticable change (both for us and the ancient Egyptians) is that the architecture in Egypt changes with the 3rd dynasty. Experimenting with mastabas turning into pyramids starts then and is prominent during the 3rd-6th dynasties, before it mostly disappears. It should be noted though that for the ancient Egyptians the whole period between dynasty 1 and 6 would have seemed mostly the same. Especially since the capital was located at Memphis the whole time.