Can someone help me understand these definitions in mechanics?

I'll just list some terms and explain what i find confusing next to them.

Inextensible String: I know that this means acceleration will be the same on both sides but i don't really know why
Light String: I think this means the string weight won't affect the mass of particles on the string but if it's light doesn't that mean it still has a mass?
Taut String: I don't really understand this
Smooth Pulley: I keep thinking this means something to do with friction but mark schemes tell me something else entirely.
Uniform Rod: I'm pretty confident this means weight won't act at the center of an object

I'll just list some terms and explain what i find confusing next to them.

Inextensible String: I know that this means acceleration will be the same on both sides but i don't really know why

Light String: I think this means the string weight won't affect the mass of particles on the string but if it's light doesn't that mean it still has a mass?

Imagine that you have a pulley where one side of the string is stretchy and the other side is taut. Can you see how this could affect the accelerations of the particles?

Copied from a document I wrote a while back...

How have you used the fact that…

The string passing over a pulley is light
The acceleration of both particles is the same

The pulley is smooth
The tension is the same on either side of the pulley

If you know why can you explain to me how a smooth pulley causes the tension to be the same

Imagine first an inextensible string without a pulley (e.g. you hold a string carrying a weight). Does it make sense to you firstly that the tension in that string is the same throughout the string?

Now a "smooth pulley" in mechanics means that the mechanism of the pulley offers no resistance to motion. It doesn't mean that the contact between the string and the pulley is smooth.

If there is no resistance to motion then a string around a pulley doesn't feel any affect from the pulley so it has constant tension just like a string without a pulley. But imagine a pulley that has heavy resistance on its left side, so the string doesn't pass smoothly anymore over the left side but does pass smoothly over the right side. The complete motion of the string isn't going to be steady anymore and this could even cause the string to bunch up a bit on one side of the pulley. You can no longer assume that the tension throughout the string is constant.