I'm assuming that by 'the modern idea of a band that plays music' you mean a small group of musicians who a) play some sort of popular music, and b) do not have a 'leader' after whom the band is named and instead have some sort of abstract name, c) are not primarily a vocal group but instead play musical instruments, and d) who usually write their own material.

It's fairly easy to identify the point where such an idea became popular/familiar to the general public: The Beatles. Biographies of the Beatles (e.g., Mark Lewisohn's 'All These Years') suggest that, when the Beatles were signed to Parlophone/EMI in 1962, the management at EMI were initially confused by the band's desire to be 'The Beatles' rather than, say, 'Johnny Lennon and the Beatles' - English record labels like EMI, at least, still largely thought about pop music in terms of 'frontman and maybe a band' until the Beatles. And not long after the Beatles hit it big worldwide, in early 1964, you get The Rolling Stones, The Kinks, The Who, The Byrds, The Doors, etc., thanks to record companies wanting to have their own Beatles.

Of course, if vocal groups count as bands, the Beatles aren't anywhere near close to being the first popular band. If vocal groups count as bands, then there's been vocal groups with abstract group names since the 1920s and 1930s at least, from the likes of The Ink Spots in the 1930s to the 1950s jazz vocal groups like The Hi-Los and The Four Freshmen, and the 1950s doo-wop groups like The Penguins or The Platters.

In terms of what influenced the Beatles' conception of the 'band', the obvious one (which Lewisohn discusses at length) is The Crickets. Buddy Holly and his band ended up being signed to two separate record labels, Brunswick and Decca, and for contractual reasons opted to release recordings under the name 'Buddy Holly' on Decca and 'The Crickets' on Brunswick. The Crickets fit almost all of the criteria above: pop music made by a group with an abstract name who play their instruments and write their own material. However, the name 'The Crickets' was something of a fiction, as they were Buddy Holly recordings in all but name.

The slightly less obvious (but still quite obvious) influence on the Beatles' conception of the band is the late 1950s/early 1960s craze for guitar-led instrumentals. The groups who recorded these instrumentals tended to have abstract band names, often associated with surfing culture (The Surfaris, The Atlantics, etc). In the UK the biggest such band was The Shadows, who were originally best known as Cliff Richard's backing band, but who became popular in their own right. And of course, the Beatles were very aware of The Shadows' career - Mark Lewisohn documents them playing Shadows songs in Hamburg.

Speaking of the surf craze of the early 1960s, The Beach Boys are the other band that have a big claim to the modern idea of the band: they were simultaneously a surf band and a vocal group, and so naturally had an abstract band name rather than a frontman + band name. They also didn't have a clear frontman/leader, wrote their own songs, and played their own instruments, and had hit singles from 1962 or so. I'm sure there's also numerous early examples of modern-style bands from before The Beatles and Beach Boys hit it big.