http://www.hdrlabs.com/sibl/archive/downloads/Tropical_Beach.zip
From the page: http://www.hdrlabs.com/sibl/archive.html
Search for “tropical beach”
Using the default Environment Texture node setting (color space: linear) it looks really dull. If I switch the node’s color space to sRGB it’s a lot more vibrant and the sunlight takes on an acceptable strength but I suspect I’m not getting much actual “dynamic range”. Should I look for another HDRI that works better when set to Linear?
Works fine for me in any colour space
Most of those are supposed to be used with an actual sunlamp (has an ibl file?). They are very heavily clipped for sun values. While I use them, I’m aware of their limitation.
But which color space looks “correct” to you? When he environment texture node is set to linear it’s like there is no real direct sunlight. When it is set to sRGB the sunlight is much better but I feel like the colors are too saturated.
So I should use it linear mode and add a sun lamp?
Yes. The image at first glance looks like linear. In sRGB it takes the appearance of a finished “fully pumped” thing - a bit over the top even. Use a mirrored ball to match sun rotation. For indoors I typically use HDRs and backplates only for camera rays, and rely on Nishita sky texture to do the actual lighting, and for this use it doesn’t bother me much that they are clipped (hi-res backplates tend to be only 8 bit anyway).
I have the HDR in question and checking, it is indeed clipped as expected. Look at the sun and reduce exposure, it doesn’t turn into a miniature 5 degree disc, but a big glob of glare, and waaaaay clipped.
what is that for?
It’s just a textfile describing to the software how the 3 images and sun(s) are to be used; background, environment, and reflection, as well as geolocation, northing, and time data. It’s not supported in Blender, and I’m not sure it’s a popular way of doing things anymore.
Could you have a look at the umbrella light on that page and tell me what’s the best way to use it or if it’s even useful in your opinion?
Well it’s up to you to decide how useful it is 
I didn’t check it but I’m expecting it to not be clipped in any way. Clipping usually happen with extremely bright light sources like the sun. At least for sunlight is where I expect it to be a problem.
To use it, I’m assuming just use it as an emission plane in your scene, possibly one sided. They’re all HDR, clipped or not, but HDRs don’t have to be equirectangular. HDR simply mean they contain more than 8 bit (256) levels of intensities per color channel (usually in the high end).
