To put it simply, the earliest poems with "song" in them were actually lyrics, specially matched to music scores for people to sing. However, in Li Bai's era, poems such as "Midnight Wu Ge" became a relatively free poetry genre. Not necessarily for singing anymore.
Poems with the word "歌" in their titles are generally ancient Yuefu poems, which are a genre of ancient poetry. According to the Records of Ritual and Music in the Book of Han, during the reign of Emperor Wu of the Han Dynasty, there was an organization called Yuefu that collected ballads from various places, compiled and formulated music scores. Later, people called the poems collected and notated by this institution Yuefu poems, or simply Yuefu. By the Tang Dynasty, although the music scores of these lyrics had long been lost, this form was passed down from generation to generation and became a poetry genre without strict meter, close to five or seven character ancient poetry.
There are many Yuefu poems since the Han and Wei dynasties titled "Song" and "Xing". Although the two have different names, there is actually no strict difference. Later, there was the integration of "singing and walking". Its syllables and meter are relatively free, and its form adopts the ancient style of five-character, seven-character, and mixed-character, and is full of changes.
The meaning of "song" remains unchanged today, and "行" means music.