What is the exact relation or difference between a color model and a color space? I find these two terms to be used interchangeably in some literature. Are they simply the same?
A color model is a method of describing a color. For example with Red, Green and Blue (RGB) elements or with Cyan, Magenta, Yellow and Black (CMYK).
A color space is the set of colors which can be displayed or reproduced in a medium (whether stored, printed or displayed). For example, sRGB is a particular set of intensities for red, green and blue and defines the colors that can be reproduced by mixing those ranges of red, green and blue.
A color model is the mathematical (or computer science) way of describing colors. It is independent from physical devices.
RGB-8 or RGB-16 are color models as well as CYMK or HSL.
A color space is the method of mapping real colors to the color model's discrete values. sRGB and AdobRGB are color spaces that both use RGB as model. But in one the color represneted by, lets say RGB(16,69,201) may be differently displayed in sRGB and AdobeRGB.
Another RGB model is describing each "channel" with a floating point rather than an integer. Both sRGB and AdobeRGB would work nicely with this mode.
Lab is a color space that agrees well with the HSL (or HSV, I am not sure) color model.
The color room is called a room because the total number of colors available inclusive of all shadeds of brightness can be shown best in some 3-dimensional corpus, rooms.
They are often used interchangeably because their separation is rather abstract. Practically you cannot really discuss color spaces without having a specific color model in mind.
Another way to add to Hermann's excellent answer would be that when the term color model is used it is used in it's mathematical form, and when the term color space is used it is usually used in the context of data that has had the color model applied to it and thus inhabits that color space.
There is no accurate term "true color" in the context of color models or color spaces, unless it's spectral data and that's not a color model or space. Since color models contantinate spectral data and attempt to create an ordered system of values, it is a safe statement to say that this ordered system of values is based on a single white point and is in itself a metameric system. So the color is no longer true, in the purest sense of that word.