Question on battery basics (I'm confused and not sure how to state my question)
You are over-thinking all this, and/or have some bad mental models. You also provide this dilemma: that it must either be "always has 9 V" or else "up to 9 V can be pulled." That's not the dilemma and I don't even know what you mean when you say "pulled."

A battery is based on some internal chemistry. There are some molecular ion potentials based upon some pretty basic physics which, luckily for us humans, means that a battery chemistry has the effect of providing a relatively stable voltage. You can even get a battery by sticking two dissimilar metal rods into different parts of a lemon, for example. (Not a "good" battery, though.)

Batteries that are designed to supply a voltage also have a "compliance current" specification for them. And sometimes, the specification will describe how well they work at several different currents: with low current situations lasting longer and high current situations lasting much shorter, over their useful lives.

When a battery is applied to a circuit (like an LED circuit), the battery's fresh chemistry setup tries to provide a roughly stable voltage to that circuit. But if the circuit requires too much current for the chemistry and physical design of the battery, then the voltage will droop. Sometimes, it will droop a whole lot, while still providing some current. Usually, batteries that are tied to a load that could use far more current than they can manage to supply, have their internal chemistry operating at such a pace that it impairs their lifetime of service, too. So it's important to stay within their designed specifications regarding the applied load.

A very simple way to visualize a chemical battery is to think of it as an ideal battery with a series resistor attached. In this way, if the circuit tries to draw too much current then this internal series resistor will "drop" some voltage before the circuit itself gets access to it. But this is just a very simple approximation. A real battery is much more complex and modeling them is an art of sorts.

But the basic idea is that a battery (or any practical voltage source) isn't perfect and has limitations. But designers depend upon the idea of a range of reasonable voltages when faced with a range of reasonable load currents. So, for example, I might design a circuit for a 9 V battery expecting no lower than 7.5 V and no more than 9.2 V from it over its lifetime of use. No, it won't ALWAYS provide 9 V. It will provide higher voltages early in its life and lower voltages later on as its chemistry gets used up. I have to decide how low I can accept, before my circuit stops working correctly. The lower I can manage, the longer the battery lasts. But the voltage also drops a lot more quickly as it gets near the end of its life, too. So I have to make a reasoned judgment about where to draw that line.