What's the difference between “Fake HDR” and real, bracketed exposure HDR?
The only difference is how broadly or narrowly you decide to define the term High Dynamic Range Imaging (HDR). Do you use the broader term as it has been historically used for over 150 years to reference techniques used to display a scene with higher dynamic range than the dynamic range of the display medium? Or do you insist on a very narrow definition that uses techniques that have only been around a couple of decades to argue that the only legitimate definition of HDR is an 8-bit tone-mapped version of a 32-bit floating point light map created by combining multiple bracketed exposures? That's pretty much it.
HDR as the term is commonly used today is only one form of High Dynamic Range Imaging (HDRI) that has been going on since at least the 1850s.
Gustave Le Gray took multiple exposures at different exposure values to create seascapes that used the bright sky from one glass plate negative and the darker sea and shore from another.
The zone system when shooting and developing and tone mapping performed in the darkroom in the mid-20th century was raised to an art form by Ansel Adams and others as they used developing times and dodging and burning of prints to lower the total dynamic range of a scene to what the photo papers they were using were capable of displaying.
In the realm of digital photography there are multiple techniques used to depict a scene with a High Dynamic Range using a medium, such as a computer monitor or print, that is not capable of as great a contrast between the brightest and darkest parts of a scene as the scene itself contains. What many people mean when they say HDR is only one such technique among many.
Though far from the only legitimate one, the most common understanding today of the term HDR is what evolved from ideas first introduced in 1993 that resulted in a mathematical theory of differently exposed pictures of the same subject matter published in 1995 by Steve Mann and Rosalind Picard. It makes a high-dynamic-range light map from multiple digital images exposed at different values using only global image operations (across the entire image). The result is often a 32-bit floating point 'image' that no monitor or printer is capable of rendering. It must then be tone mapped by reducing overall contrast while preserving local contrast to fit into the dynamic range of the display medium. This often leads to artifacts in the transitions between areas of high luminance values and areas of low luminance values. (Even when you open a 12-bit or 14-bit 'raw' file in your photo application on the computer, what you see on the screen is an 8-bit rendering of the demosaiced raw file, not the actual monochromatic Bayer-filtered 14-bit file. As you change the settings and sliders the 'raw' data is remapped and rendered again in 8 bits per color channel).
When the techniques outlined by Mann and Picard were first applied in mainstream consumer level imaging applications, those applications usually required the images used to be in jpeg format. A little later on, if you wanted to get real exotic, you might find a program that let you use TIFFs. Often users would take a single raw file, create a series of jpegs from the single file with something like -2, 0, +2 exposure/brightness differences and then combine them using the HDR program. Even a 12-bit raw file can contain as much dynamic range as a -2, 0, +2 series of jpegs. A 14-bit raw file can contain the equivalent information as that in a -3, 0, +3 series of jpegs. Only fairly recently have most HDR applications based on creating floating point lightmaps allowed the use of raw file data as their starting point.
In the broadest use of the terms HDR (or HDRI), other processes that do not involve 32-bit luminance maps and the necessity of tone mapping are also included. Combining different areas of different exposures of the same scene, whether via a physical 'cut & paste' as Le Gray did over 150 years ago or via modern digital imaging applications that use layers, is one way. Other techniques, such as Exposure Fusion or Digital Blending digitally perform global adjustments in a way that doesn't require the same type of tone mapping that a 32-bit floating point light map does. As mentioned earlier, many of the techniques used in the darkroom to produce prints from exposed film in the 20th century were a means of displaying scenes with a very wide dynamic range using photographic paper that had the capability of a lower dynamic range than the negative film used to capture the scene. The same is true of these varied digital techniques.
Even converting a 14-bit raw file, where the data for each pixel only has a luminance value but no real colors, and using demosaicing algorithms to interpolate an 8-bit per color channel red, green, and blue color value for each pixel based on the different luminance values of adjacent pixels that are filtered using a Bayer mask with alternating patterns of red, green, and blue can be considered HDRI, especially when irregularly shaped tone curves are applied to the resulting RGB values.