Astronomers find the edge of the Milky Way

To use an analogy, to add to everybody else: it's like rings on a tree stump. The innermost part of the stump is the oldest; the outer the youngest. Earth is on one of those in-between rings, neither the oldest nor the newest - doesn't matter which of the in-betweens, to be honest.

Suppose now that you're an ant on the middle ring of that tree stump. No matter which way you're looking from Earth's middle-ring, either the rings will get gradually older and then younger with increasing distance (if you're looking towards the center-ish), or the rings will get strictly younger (if you're looking away from the center-ish).

This analogy obviously breaks down if you delve into details but that should give a better intuition to what's going on.

I actually am not following what the ambiguity is - stars farther out from the center are younger, no?

When does "starting in the center" mean anything besides "starting in the center"?

The earth is not the center of the galaxy

Poorly phrased. The most recent stars are on the edges. The inner stars were first, hence the “working outwards”.

Looking from Earth at the stars closer to the center of a galaxy, they are found to be older. Looking from Earth at the stars closer to the edge of a galaxy, they are found to be younger.

Earth isn't relevant. The stars at the center of the galaxy developed first, and development proceeded from the inside out, so the youngest stars are on the edge ... then they get older from there on out, as the stars beyond the edge broke away from the galaxy. The bottom of the age U is the location of the formative edge.

From Earth (where astronomers are) looking "out"ward (away from the bulge).

Try: "the farther out [from the center] astronomers look"

It is beyond obvious what they mean.

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I prefer that feeling much more than the modern sense from social media where everyone is abnormally important