There's nothing inexplicable about that photo. It's got a few uncommon features, but nothing unknown. In the image below, I've drastically distorted the brightness curve to make things more obvious:
- Primary arc. This is the classic "internal reflection" rainbow that you learn about in school.
- Secondary arc. This one forms when light is reflected twice within a raindrop, broadening the bow and reversing the order of the colors.
- Supernumerary arcs. These are formed by the wave nature of light: light, even of a single color, spreads out as it passes through a raindrop. The arcs are interference patterns formed by the spread-out light from many small raindrops, and because of this spreading, they're usually pastel colors rather than the single-frequency colors of the main arcs. The less variation there is in raindrop size, the sharper the arcs are; these are especially sharp.
- Supernumerary arcs on the secondary bow. These are rarely seen, because they're even more sensitive to variations in raindrop size than the primary arc's supernumeraries.
- Possible fifth-order bow. This faint green patch is in the right place to be the green stripe from light reflecting five times as it passes through a raindrop. Fifth-order bows are almost never seen because of how faint they are, and are photographed even less often, so this may just be an image artifact.
- Primary glow. Only the light passing through the edges of a raindrop forms the rainbow. Most of the light passing through the middle just goes straight through to form the zero-order glow (not pictured), but some reflects back to form a glow in the middle of the rainbow.
- Alexander's dark band. The light making up the rainbows has to come from somewhere, and this band is where it comes from.
- Reflected rainbow. This isn't a reflection of the primary arc (rainbows aren't real objects, so they don't have reflections), it's the rainbow you'd see if your position had been reflected across the plane of the water.
All of these have been known to science for quite some time. The primary arc was the first one to be explained, by Persian astronomers in the 1200s. Supernumerary arcs were the last one to be explained, by Thomas Young in 1803.
The "third rainbow" the BBC is talking about is probably the first few supernumerary arcs of the primary arc. Supernumerary arcs have the same color ordering as the arc they're attached to, but because the brightness and width of the arcs varies, and because they're usually pastel colors rather than spectral colors, it can give the appearance of a reversed spectrum.
The "third rainbow" isn't a twinned rainbow, because the second arc of a twinned rainbow (caused by two rainstorms at different distances with different raindrop characteristics) wouldn't be concentric with the first arc.