I think of it as the Terry Richardson look. He's been going to the well on it every day now for several years. Earning big bucks with it too, shooting major models and stars for high-prestiege clients. Many of his subjects are super-fit, trim and are model-y types, entertainers, and so on. years ago, Weegee made a career with on-camera flash for the most part. It has its unique attributes, especially in B&W, where it looks better, I think, than it does in color. Again though, we need to differentiate between on-camera FILL lighting that does little more than lighten the shadows and eye sockets, and using on-camera flash that's so bright it acts as the MAIN light source. Big difference.
What's odd is that 25-30 years ago, the OP's first shot might have been viewed as "good", and "trendy", because on-camera, TTL balanced flash was the hot, new thing. Then came the "portable softbox with square catchlights in the eyeglasses" look. The visual language now has shifted to the point where some people just absolutely think flash MUST come from off-axis angles, and nowhere else,in all cases, which is not a position I agree with. The decade old internet joke, "Needs fill flash," C&C comment has now been supplanted by militant assertions of "Needs Off-camera Flash." No offense is meant to either the OP or to those who responded to her--I'm just commenting on how trends come and go, and expectations shift and change...and sometimes, go BACK to the old ways...