Do you have any pictures of the first installation?

It looks a lot like this (the photo with the tile grouted):

Here is a link that might be useful: Handmade Tiles

In that case, he set the tiles slightly skewed in various directions, with varying grout gaps to give the illusion of being of slightly different sizes,

Then he "raked" the joints (scraped the grout out a bit) so the grout is not even with the top edge of the tiles.

I don't think a deliberately bad install job fools anyone to thinking you are using handmade tile.

"It looks a lot like this (the photo with the tile grouted)."

Um, no it doesn't. Not to anyone who knows tile. Ditto what holly said.

You liked the previous install so I'm sorry to criticize, but with handmade tile you have irregular rectangles with an attempt at regular spacing. Your previous install was regular rectangles with an attempt at random irregular spacing.

In the image you link to, there is lippage, a cupping of the tiles that occurs during firing. You can't mimic that. Even an untrained eye will spot machine manufactured sameness of the tiles, no matter how odd the installation.

If you're installing perfect tile, do it right. If you want handmade tile, install a few pieces as a feature in your bs.

Sounds like I didn't do justice to my previous tile installer. Here I was assuming, and complaining about, that my tiles were just a bad tiling job while he was trying to achieve a hand-made tile look :)
I'd have the tiles installed the way they are intended.