In thinking Being, Heidegger differentiates between
- The first beginning
- The other beginning
The first beginning starts with the ancient Greeks. They ask the question of Being (Sein), but in a specific form, namely what a being (Seiende) in essence is. In Sein und Zeit, this is called the present-at-hand (Vorhandenheit) mode. Later he calls this the beingness (Seiendheit) of being. This way of asking the question assumes the Being of being is itself a kind of a being (Seiende), namely fundamental and eternal: an essence, an idea (Plato), an ousia/morphe (Aristotle), a representation (Kant), will to power (Nietzsche) etc.
The other beginning starts with Heidegger (sort of, he usually says he's making way for it, preparing for it). Rather than (merely) thinking Being at the being present at hand, he questions Being as Being, Being as such, Being itself the clearing of Being.
The first beginning obscures the real question of Being. So they (and all other philosophers until Heidegger) are 'thinkers of Being' only in a certain limited way. The ancient Greek thinkers had an experience or insight into Being, but by asking about it in this way, this insight was obscured and soon forgotton.
Plato and especially Aristotle are the ones who pushed to formalisation of this way of thinking and are therefore presented by Heidegger as the origin of the forgetting of Being. In the presocratic philosophers (Heraclitus, Parmenides) this formalisation is less developed, and usually Heidegger hopes to find traces there of the original insight into Being, to help him with making way for the other beginning.