Unicase is another moniker; right or wrong, it's different enough to get noticed, which I suppose is the whole point...

It makes me feel unbalanced; I don't like it.

As long as a good unicase font is used or a good lettering artist is hired it’s a great idea. Just don’t leave it up to some hack who thinks that randomly tweaking bits of letters is logo design.

I agree with the above comments. I think it's an interesting concept, I think it definitely makes you think "what the...?" I think it makes sense when a company is unlike it's competitors or wants to be seen as mold-breaking or different.

For what it's worth, here is a concept I'm working on for a client that is modernizing, and unlike any in it's sector. They want to convey that, yet look serious, and corporate-clean.

I'm doing it with a san-serif. Screen resolution is making some of the vertical strokes appear different, I can assure you the I's are the same widths, and are the same width as the middle crossbar on the lowercase e.

Feel free to critique my kerning and character widths. I'm thinking the P could be a little wider.

…from a purists standpoint it's awful…

Not at all.
Unicase has a good pedigree, going back to uncials.
It was formalized intellectually by Bradbury Thompson, a well-respected designer, as Alphabet 26.

The sample you show is set in Filosofia, one of many types that has a unicase option.
The modern/didone style has proved to be quite suitable for unicase (I included it in the display cut of Scotch Modern).

My unicase type system: Panoptica

Here's another example I've seen around the East Village.

Those letters seem awfully light for use as a logo. As you’ve already found out those strokes won’t render evenly on screen, and you’ll have similar problems any time it gets printed small on laser or inkjet printers. That said I think that your design is strong but the spaces between P and I and N and T are too wide. The S might need to be a little wider to balance it against the wide minuscules and the big air gap to the left of it.

The Winebar is a classic example of the hack category. Someone simply scaled capitals down to the x-height and didn't notice or care that the weights are now mismatched.

Agreed - the "WINEBAR" logotype is awful. Especially bad choice to use Bodoni, a face that is predicated on equal proportions in the variation of stroke width. I feel unicase type is tough to execute with any face that has that much variation. Lowercase was not designed to have the same x-height as uppercase so you can't just mash them together and call it a day - requires some finesse. Plus some balance is needed perhaps equal or close to equal number of upper v. lower characters would help keep the balance.

The NeXT logo, designed by Paul Rand in 1986 for Steve Jobs...

If I remember the story correctly, he told Steve Jobs that he would present only one concept -- take it or leave it -- and his fee would be $100,000 (although that amount covered the entire identity system, not just the logo itself).