Vālmīki's depiction as a Robber
It looks like Tagore's Vālmīki-Praṭibhā is partly based on Kṛttivāsa Rāmāyaṇa. In the opening chapters of his Rāmāyaṇa, Kṛttivāsa narrates how Ratnākara, son of sage Cyavana, used to be a dacoit before transforming himself into Vālmīki Ṛṣi by meditating on the Rāma-mahā-mantra.
From Ādi-kāṇḍa Ch. 1 (Birth of four aṁśas of Nārāyaṇa):
Brahmā with a smile, asked Trilocana Śiva, to name a great sinner on earth. Śiva then asked Brahmā to listen attentively to him; He said, "On the madhya-patha (mid-way) there is a great sinner, who can achieve deliverance by the reciting of the name of Rāma. Go and give the Rāma-mantra to him which would remove his sins. Nārada then thought of the identity of such (a sinner) person, who was none else than Ratnākara the son of sage Cyavana. He had been a dacoit, and lived on forcibly snatching others belongings and was a great sinner living in a forest.
Both Nārada and Brahmā then went (to the forest where Ratnākara lived) disguised as ascetics. As luck would have it, Ratnākara could not loot any one because no one passed that way on that day. He then climbed high in a tree and found Nārada and Brahmā coming together. Concealing himself (in the forest) he thought that he could snatch away the robes of the two ascetics.
From Ādi-kāṇḍa Ch. 3 (Ratnākara named as Vālmīki by Brahmā, and the boon of creating Rāmāyaṇa):
Brahmā spoke to the sage, "Up to now, you were known as Ratnākara, but henceforth you shall be known as Vālmīki. The name of Rāma has purified you, therefore, you will create his Rāmāyaṇa in seven kāṇḍas." Vālmīki, then spoke with humility, "I am quite an illiterate person having no knowledge of Kāvyas and Chandas; how can I create literature?" Brahmā, on hearing the words of Vālmīki spoke to him, "Sarasvatī resides in your tongue and as such you will be able to create the Kāvya quite easily, whatever verses you will create, will be devoted to Rāma in the universe." Awarding the boon to Vālmīki, Brahmā went to his abode; Vālmīki was immensely pleased (by his achievement)...
According to Tagore's own summary of the play:
 Valmiki Pratibha means the genius of Valmiki. The plot is based on the story of Valmiki, the robber chief, being moved to pity and breaking out into a metrical lament on witnessing the grief of one of a pair of cranes whose mate was killed by a hunter. In the metre which so came to him he afterwards composed his Ramayana. Tr.
However, according to the Wikipedia article of the play, Vālmīki had a change of heart after capturing a young girl to be used in a sacrifice, so, obviously, this account is very different from what's told in Kṛttivāsa Rāmāyaṇa.
One night, Valmiki, the robber chief, and his men captured a young girl to be sacrificed before Kali, goddess of death. As Valmiki approached to behead the girl, her cries melt the robber chief's heart and she was released. Later Valmiki's men found that their leader was taking no more pleasure in bloodshed. Considering it a shameful cowardliness, they abandoned him.
How does Vālmīki Rāmāyaṇa describe Vālmīki?
First off, Uttara-kāṇḍa of Vālmīki Rāmāyaṇa says that Vālmīki is the son of Pracetā (not Cyavana). And Vālmīki in his testimony (of Sītā's chastity) to Rāma says that uttering a lie didn't arise even in his thoughts, ever. So he could not have been a robber.
I tell you the truth, O Rāma, that these irrepressible twin brothers are your sons, O descendant of Raghu. I am the tenth son of Pracetā, so, far from speaking untruth it does not even spring up in my mind. I therefore know the truth that these twins are your sons. I have performed austere penances for many thousand years; I now swear before you, that if this Maithili is found touched by any sin I shall not reap the fruit of my ascetic observances extending over many thousand years.
Vālmīki as depicted in Skanda Purāṇa
As noted in this answer and this & the next parts of Skanda Purāṇa Vol. VI Ch. 124, Vālmīki was born into a brāhmaṇa family and was previously known as Lohajaṅgha (belonging to the family of Māṇḍavya). He resorted to stealing things following a famine. A group of sages later rechristen him as Vālmīki after he's transformed himself into a ṛṣi.
1-2. Then there is another excellent Tīrtha named Mukhāra where excellent Brāhmaṇa-sages had contact with a thief. That thief attained Siddhi, thanks to their power. Later he composed the epic Rāmāyaṇa and became well-known as Vālmīki.
3. Formerly, there was a Brāhmaṇa named Lohajaṅgha in Camatkārapura. He hailed from the family of Māṇḍavya. He was devoted to his father and mother.
13. Achieving his purpose thus, everyday by the acts of a thief, he maintained his family through thievery.
14. When the famine passed off and the time of plenty arrived, he never did any personal work. Forsaking his Brāhmaṇical pursuits, he continued the practice of a thief.
The sages said:
85. Since you achieved great Siddhi seated within a Valmīka (an anthill), you will become well-known in the world as Vālmīki.
Of all the above, Vālmīki Rāmāyaṇa's depiction is probably the most authentic simply because it's considered the oldest of the two Itihāsas and the Purāṇas.