Is a real Grayscale image always 8 bit per pixel indexed? I tested several Grayscale images downloaded from the Internet. Some of them are 8bpp indexed while others are 24bpp (true colored). Looks like different Grayscale images have different depths.

For instance, this online application converts a color image to Grayscale. But, when I check its bit depth, I see that bit depth is not changed.

I think, Grayscales with 24bpp are pseudo-grayscales. That is, they are the average of R, G and B components. I think, real Grayscales are always 8 bit per pixel indexed.

Is my notion correct?

If not, please, provide me some links that I can study.
I don't think that it is that easy. Grayscale images do have certain benefits when you think of imaging experiments. When the measurement is squeezed into eight bits, even though the measurement is much more precise, then there is a loss of information due to the quantization. See the DICOM format for example that is widely used in medical imaging. It is very common there that the information is encoded in 12 bit, or even more.