Both "every time you flip a coin, you create a pair of parallel universes" and "That everything that could ever have happened did in fact happen" are both fairly big misinterpretations of many worlds theory – both in fact (there's no real connection between quantum measurement process and a coin flip or other macro-scale outcomes), and in reasoning (the theory does not posit many worlds, they came out organically from the math).

Everettians and non-Everettians agree that
1) The state of the universe is fully described by a set of quantum fields
2) The quantum fields will change over time according to the Schrödinger equation.

So far, this is analogous to how classical physics is set up, but in classical physics the "state" is a bunch of particles with positions and momentums, and the "change over time" is F=ma (for all forces).

Non-Everettians would say those two rules alone do not explain experimental results. We need extra rules, namely:
3) When a "measurement" occurs, the state of the quantum field collapses into a single definite value, and
4) The probability of the outcome of that measurement is given by the Born rule

The main issues with non-Everettian interpretations are that 1) The definition of a measurement is vague, and 2) you need some theory of why the collapse occurs, and what exactly is happening as the measurement is being taken (there are many options here).

Everettians instead ask, what would happen if we just... didn't have those extra rules?

If so, the Schödinger equation on its own predicts the following evolution of quantum state during an experiment:

To begin we have an "uncollapsed" quantum state. Say, an electron in a superposition of spin-up and spin-down, not entangled with anything else

During the experiment, the electron becomes entangled with other particles in the measurement apparatus

By the time the measurement is complete, we have full entanglement, the new quantum state is a superposition of "spin-up and a small current induced in the top plate" and "spin-down and a small current induced in the bottom plate"

When the experimenter reads the result, we have even more entanglement into a superposition of "spin-up and the experimenter sees it's spin up" and "spin-down and the experimenter sees it's spin down"

There are cascading interactions with every particle in the universe until the whole universe is entangled into a superposition of "spin-up and the universe is as it would be if the particle were spin-up" and "spin-down and the universe is as it would be if the particle were spin-down"

At this point an Everettian would say: this is exactly consistent with what non-Everettians were trying to show with their extra rules! The quantum field has evolved into a superposition of two universes, and the Schrödinger equation predicts those two universes never interact with each other... so there's really no evidence that the Schrödinger equation ever needs to not hold true. Quantum collapse happens on its own when you just run the Schrödinger equation forward, and the extra universes are just an unobservable byproduct.

The main issues with the Everettian interpretation are epistemic: I don't know what it should feel like to be in a superposition of two non-interacting universes with different states of knowledge, but to be an Everettian, you need to believe it feels like just being in one of them. Depending on your philosophy of consciousness, it's anywhere from trivial to impossible to prove that.