What is the relationship between Tagore's poems and his song lyrics?

Question

Rabindranath Tagore composed and wrote the lyrics for approximately 2,200 songs, collectively known as Rabindra Sangeet, "Rabindra music". These songs remain immensely popular in India and Bangladesh; each country has a Tagore composition as its national anthem. Outside South Asia, Tagore is best known as a poet, but on the subcontinent, his songs are easily the most widely circulated of his works.

Yet the status of Tagore's song lyrics vis-à-vis his poetry is hard to pin down. His poetry and songs are often treated as two separate entities. For example, TagoreWeb, the online portal that presents all Tagore's works, maintains two separate headings for "Verses" and "Songs". The English Wikipedia entry on Tagore also has separate sections for "Poetry" and "Songs". The latter includes a throwaway phrase that suggests his songs are categorically different from his poems:

For Bengalis, the songs' appeal, stemming from the combination of emotive strength and beauty described as surpassing even Tagore's poetry, was such that the Modern Review observed that "[t]here is in Bengal no cultured home where Rabindranath's songs are not sung or at least attempted to be sung....Even illiterate villagers sing his songs". [emphasis added]

But it is not clear whether the added "emotive strength and beauty" derive from the music's enhancing the song text, or whether the lyrics themselves are somehow different from, and more immediately appealing than, his poems.

The Bengali Wikipedia entry is equally enigmatic about the relationship between Tagore's lyrics and his poetry:

রবীন্দ্রনাথের গান তার সাহিত্যের সঙ্গেও অঙ্গাঙ্গীভাবে জড়িত – তার বহু কবিতা যেমন গানে রূপান্তরিত হয়েছে, তেমনই তার উপন্যাস, গল্প বা নাটকেও বিশিষ্ট ভূমিকা নিয়েছে তার গান।

Rabindranath's songs are intimately linked to his literary works—just as many of his poems have been transformed into songs, so also his songs have played a prominent role in his novels, stories, and drama.

The entry does not elaborate any further on the relationship between Tagore's songs and his literary output. However, the statement that "many of his poems have been transformed into songs" does indicate that there is at least an overlap between Tagore's poetry and his song lyrics.

In the best essay on Tagore that I have read, Amartya Sen also comments on this overlap:

Tagore’s poetry, which often takes the form of songs in an innovative style of lyrical singing, called Rabindrasangeet, has transformed popular Bengali music with its particular combination of reflective language and compatible tunes.

It is not clear, however, exactly how Tagore's song lyrics fit in with his poetry as a whole. For example, looking through the Bengali Gitanjali (1910), one comes across some poems whose musical settings are well-known. One example is Poem 8, "আজ ধানের খেতে রৌদ্রছায়ায় / aaj dhaaner khete raudrachhayaay", "Today, in the dappled sun and shadow of the rice field"). This poem is included under both the "Verse" and the "Songs" sections of Tagoreweb, with the line breaks being the only apparent difference. Other poems in the Bengali Gitanjali haven't been set to music at all, or at least, not by Tagore (e.g., Poem 1, "আমার মাথা নত করে দাও / aamaar maathaa nata kare daao", "Compel my head to bow").

The same can be observed of the poems Tagore translated himself. Analyzing the English Gitanjali (1912), Partha Pratim Ray and B.K. Sen note that 42 of the 103 translations are of poems that Tagore had given a musical setting (see Table 10). That leaves 61 of the translations to be presumably of poems that Tagore had not set to music.

The question, then, is: what is the relationship between Tagore's song lyrics and his poems? To be more specific:

• It is clear that Tagore set some of his poems to music. On what basis did he choose these poems? Are they stylistically or in some other way clearly distinguishable from those he did not set to music?
• Are the lyrics of Rabindra Sangeet a subset of Tagore's poems? That is to say, is the corpus of Rabindra Sangeet merely those poems Tagore chose to set to music? Or are there songs Tagore composed whose lyrics are not considered poems—i.e., which were published in songbooks rather than in collections of poetry? If so, what distinguishes those lyrics stylistically from his poetry proper?
• Tagore's plays incorporate song as part of the action. Some of his best-known songs are in fact from his plays. Were the included songs written specifically for the plays, or were they poems or pre-existing songs of his that he repurposed for the plays?

Notes

1. Standard disclaimer: all translations from Bengali above are my own, and my Bengali is dreadful, so rely on them AYOR
2. Transliterations from Bengali follow the iTrans scheme.