Can someone please clarify what exactly is meant by magnification? From what I've seen there appear to be 3 different definitions of magnification:

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*The angular magnification of a telescope or microscope compares the ANGULAR size of the image to the angular size of the object. This one makes the most sense because it will make the image actually look bigger at the eye which is what I associate the most with the word magnification. Unfortunately this can only be achieved with a system that has multiple lenses, which leads us to...


*The linear magnification, which compares the height of the object and image. I fail to see the utility of this because it seems to me that if the angular size doesn't change then a change in height doesn't matter because it won't look any bigger.
But, as said above, if you only have a single lens imaging device, such as a magnifying glass, or the lens in the eye, the latter definition is the only type of magnification that can happen. This is because:

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*We require that the image be defined by the point at which ALL paraxial rays through the lens intersect.

*We know that a ray going through the centre will be undeflected.

It therefore follows that the image will have to be somewhere along the line going from the object point through the centre of the lens. Thus, the angular size of the image must be the same as the angular size of the object; this follows from basic geometry of opposite angles between two intersecting lines. There's a diagram below to explain this a little clearer:
Surely from this it would logically follow that there cannot exist a formula for the ANGULAR magnification of a magnifying glass? Well this leads to the third and most confusing definition of all:


*The angular magnification of a magnifying glass (yes, contrary to everything I've said above, this is a thing) is given by the formula $M = 1+\frac{PP}{f}$ for an object at the focal point or just $M = \frac{PP}{f}$ for the image being at PP, where PP is the Punctum Proximum (or near point in English). A nice derivation is seen in this video if you're unfamiliar with these formulae: https://www.youtube.com/watch?v=U5pTpXUM7EQ&ab_channel=AndreyK.

This definition rather sneakily allows angular magnification from a single lens by taking the ratio of the angular size of the image when using the lens to the angular size of the image without using the lens, crucially when the object is not at the same place for each case. For the first formula for example, $M = 1+\frac{PP}{f}$, this is given by:
M = (Angular size of image produced by object at the focal point, when using the lens)/(Angular size of image produced by object at the near point, no longer at the focal point, without using the lens)
So what actually am I asking? A few things:

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*Why do we care about linear magnification if it doesn't actually make the image look bigger?

*Quite often the word 'magnification' is used without specifying which of the three definitions above is being referred to. For example, I have a problem where I am asked to find a formula for the magnification of a lens in terms of $f$ and $x_o$ from Newton's formula $x_o x_i = f^2$. What does this mean? Is there a default definition to use or do I just need to learn to be able to know from context?

Thanks in advance!
EDIT: for my second question, I think I need definition 2 as it's the only one that uses both $f$ and $x_o$ in the expression for magnification. But unsure of how I was supposed to know this without doing the question first.
 A: First, physics is about the phenomena of the real world. Get a magnifying glass and observe that it actually produces a magnified image as far as your eye is concerned. There is nothing sneaky about capturing this fact in an approximate formula.
So, how to make this quantitative. Indeed, the point is that the single lens magnifier allows you to change the geometry, while still focusing the image on your retina. The convention is that healthy eyes can focus at 25 cm (I learned 10 inches back in the day). The magnification is, by convention, taken to be the linear magnification of the image on a human retina, relative to the image produced with the object 25 cm from the eye. Note that is is not linear magnification relative to the object itself.
A: To your question 1: one uses the linear magnification for projectors, the image on the screen is a linear magnification of the object on the film, angular magnification there makes no sense since the angle you see the picture on the screen depends on where you sit.
