The short answer is that no, they didn't have specific names for the different stages of their history. In fact, in many cases, they didn't even record the names of all of their Pharaohs.
The Story of Sinuhe dates from the Twelfth Dynasty (the oldest surviving copy being from the reign of Amenemhat III) - which was, as you observed in the question, long after the pyramids had been built. The author, Mika Waltari, is, perhaps using the term "age of the pyramids" simply as a poetic way of saying "a very long time ago".
Indeed, the Ancient Egyptians may well have used such colloquial phrases, but no evidence of that use has yet been discovered.
In his book, Royal Annals of Ancient Egypt, the Egyptologist Toby Wilkinson notes that:
Prior to the creation of the Turin Canon in the Nineteenth Dynasty, there is little evidence that the Egyptians regarded the recording of history as an objective exercise. Lists of kings were compiled for religious and political reasons: to honour the cults of revered ancestors and to stress the legitimacy of the reigning king as latest in a long line of rulers stretching back in an unbroken succession to the time of the gods.
- [Wilkinson, Toby A. H., Royal Annals of Ancient Egypt, Wiley, 2000,
p62], my emphasis.
So there would have been no need for Ancient Egyptians in earlier periods to divide their history into 'periods' or dynasties'. This is also important, since the king lists mentioned here provided the main framework upon which later authors would base their divisions of Egyptian history into 'periods' or 'dynasties'.
It seems certain that there must have existed some royal archive, or something similar, which is now lost but which recorded lists and some details of past kings. This archive is what must have been used from the Nineteenth Dynasty to create the more detailed lists which do survive (at least in parts).
However, since we still do not know precisely why those later lists were compiled (including Manetho's Aegyptiaca) we should still be cautious about inferring too much about the sources used or their content when using them.
The King Lists
Wilkinson mentions the following king lists which survive (along with Manetho's 'dynasty' under which they were created)
- The cylinder seal from the tomb of First Dynasty king Den: Lists
all the (known) 1st dynasty kings from Narmer to Den by their Horus
- Palermo stone (5th dynasty): Broken and incomplete.
- Giza King List (6th dynasty): known to be very selective.
- South Saqqara Stone (6th dynasty): Also very selective.
- Karnak King List (18th dynasty): Very selective.
- Abydos King List of Seti I (19th dynasty): Very detailed list,
although it completely omits what we now call the First Intermediate
- Abydos King List of Ramses II (19th dynasty): Very selective.
- Saqqara King List (19th dynasty): Another very detailed list,
but omits most of the 1st dynasty Pharaohs we know of from other king lists.
- Turin Canon (19th dynasty): Very fragmentary, but may be the
most complete surviving king list. This list also appears to include
the Hyksos rules.
Plus, of course, the list compiled by Manetho that you mentioned in the question.
Apart from Manetho, these king lists do not attempt to divide the kings into periods or dynasties. Their purpose appears to have been to show an unbroken line of (legitimate) kings, who ruled according to mꜣꜥt, or Maat, and so - presumably - legitimise the current rulers.
Funerary Monuments and Tombs
In addition to the king lists mentioned above, we often get a lot of information about individual Pharaohs and their reigns from inscriptions on statues, funerary monuments and tombs. However, these are usually confined to recording the deceased Pharaoh's life and achievements in the best possible light. As pboss3010 noted in the comment, when referring to the deceased Pharaoh (as in almost any case where a text refers to a deceased person, whether royal or otherwise), these inscriptions will include the epithet "justified".
Once again, we do not get references to Egyptian history, so as we might expect, we don't see mentions of periods or dynasties here either.