Someone recently asked about a Koehler & Hinrichs Pool Table with Only 4 Corner Pockets about a month ago (but he too didn't share a photo).
Your question reminded me of Earl Strickland. He always talks about how he hates side pockets. Like, literally, always.
The story goes that Earl is always saying the only way anyone ever beats him is because of the side pockets. In fact, I've read stories that say Willie Mosconi had also been known for saying:
Jump cues and side pockets are for wimps.
Diamond made one some years back and apparently Earl Strickland and Darren Appleton played some matches on it over the weekend of May 6-8, 2016 in Akron, OH. They played call-shot 10 ball, and raced to 75, with $10,000 on the line.
I came across this photo of a Connelly Billiards 4-pocket pool table too, but upon closer inspection, it appears photo-shopped.
Here's a D. Harris and Son snooker table with only corner pockets and no side pockets. It was showcased on the snookerheritage.co.uk website. They got the info from Gerard Kearney of Irish Billiard (Snooker) Company of Long & Co. based in Cork. Here's the 4-pocket snooker table and the details:
Here's what they had to say about the 4-pocket tables (edited slightly for clarity):
Mr. Kearney worked on this D. Harris snooker table in the 1980s. It was 8' x 4' and only has corner pockets! Quite how anyone plays any form of Billiards on this table is something of an enigma. Peter Ainsworth, a notable snooker historian had researched this question in the past. In his research, he became aware of some American 4-pocket billiard tables being produced. He noted that some of the American snooker tables were made in the period from 1863 to 1869, at a size of 6' x 12' with only four pockets, followed by some made from 1869 to 1873 at sizes of 5 ½' x 11', again with only 4 side pockets. Then from about 1873, full carom tables (without any pockets) were being made at 5' x 10'. So from this, Peter Ainsworth feels that the 4-pocket D. Harris snooker tables must be pre-1873 when the fashion for 4 pocket tables died out in America.
I also found some history from The Chicago Billiard Museum:
Four-pocket pool tables were actually the standard for many games back in the middle 1800's. In April 1864 the first technically "public" competition of 3 ball caroms was played on a 6x12' four-pocket billiard table.
That was from 2014, and the founder of the Chicago Billiard Museum has since passed away.
Also, there's a pool table variant with only two pockets at one end, I think they're used to play a variant of one-pocket called "Corners".
See: Rules for "Corners" Game
Here's one of the two-pocket billiard tables:
These two-pocket pool tables were apparently common in the 1920s for this one-pocket variant.
If you come across any more info that what I've shared here, please to share it with us. Also, I sincerely hope you can get some clear photos of your 4-corner-pocket pool table to us. I'd love to see yours. Try uploading photos here... and if you encounter issues, let me know and I'll get an email address to you.