That image shows 2 x 3.5 mm plugs, whereas the second BA socket is not 3.5mm, whereas the first one is. The single and two bands indicate a TS and TRS plug. TRS (tip, ring, sleeve) has left, right and ground, and is what is needed for almost all 3.5 mm phones around. Some propriety system have another band, to create TRRS, tip ring ring sleeve and that can come on smartphone headphones, for example where there is a mic involved. There are many different types, but those non standard headphones will benefit from that adapter in single jack mode.

In other words if you have a standard 3.5mm headphone, 2 black bands, that's TRS and you do not need an adapter.