The basic linguistic history is actually pretty simple, but for whatever reason, it is not understood widely enough.
We can start the journey loosely around 1000 BCE in North-Western India/Eastern Pakistan for our purpose. The language here was what linguists term Old Indo-Aryan. Its literary variant is now called (late) Vedic Sanskrit. Writing was most likely not used in this culture. There was already a vast body of literature, but it was oral.
In the next 500-600 years or so, the language had already reached Bengal to the East and Deccan to the South. However, the spoken dialects far from the North-Western heartland had diverged quite a bit from the literary language of the Vedic literature. This is what linguists term early Middle Indo-Aryan. However, the scholarly as well as orthodox priestly classes were still trying to maintain the old standards of the Vedas in the literary and formal language. However, not all grammatical intricacies of the Vedic could be maintained, and the vocabulary had also evolved, but the sound-system was essentially preserved in the formal speech. A standardization of this formal language takes place at this point in the hands of Pāṇini et al, which would remain remarkably stable for the next 2 millenia, and acquire the name Sanskrit a few centuries later. But to Pāṇini it is still simply bhāṣā (=speech < bhāṣ- = to speak) as opposed to chandas (~sacred text), i.e. the Vedic texts. The informal colloquial dialects (i.e. early Middle IA), in the meantime, already feel quite different, especially in the East. They start to find two important formal uses, viz. in heterodox religious movements (e.g. Buddhism), and slightly later in imperial edicts to pass instructions to the masses. These latter ones are the oldest proven samples of writing in Indo-Aryan languages. These various early MIA languages/dialects are now known as Pali (Pali's exact history is a bit more convoluted though) and Ashokan Prakrits (the ones used in the edicts of Emperor Ashoka), but I am not aware of any contemporary names for them.
Let's fast forward another 1000 years (i.e. 500 CE). The spoken Middle Indo-Aryan dialects have been evolving naturally, and diverging across Northern subcontinent. Certain idealized versions of these languages are being used in poetry and drama already, and as a medium of literature they are called Prakrit (prākṛta in Sanskrit means "natural"). The Old Indo-Aryan formal language, as standardized by Pāṇini, etc. a millenium ago is also being used vigorously in literature and polite educated speech, and is called Sanskrit (saṃskṛta in Sanskrit means "refined/constructed") in opposition to the Prakrits. In essence, however, this Sanskrit is same as what Pāṇini had called bhāṣā (=speech). This is also what we now call Classical Sanskrit, in opposition to Vedic Sanskrit, i.e. what Pāṇini had called chandas.
Another 600-700 years later (i.e. roughly coinciding with the beginning of Turkic/Afghan conquests), the colloquial Indo-Aryan dialects have evolved further into what are considered the oldest forms of the New Indo-Aryan languages like Hindi, etc.
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So, Prakrit, properly speaking should refer to the form of Middle Indo-Aryan spoken roughly in the 1st millenium CE, especially in its first half.
