Stool can lodge is a prolapsed rectum as it is stretched out, often forming a pocket. I never (knowingly) had it protrude from the rectum.

The way to tell the difference, per Cleveland Clinic:

Hemorrhoids
Swollen veins in the anal canal and around the rectum. Symptoms include bleeding, itching, pain, swelling, and bulging veins.

Rectal prolapse
A condition where the rectum slips down into the anus. Symptoms include bleeding, a bulge outside the anus, itching, and the feeling of sitting on a ball.

Here are some other differences between hemorrhoids and rectal prolapse:

Location
Hemorrhoids are located near the anal opening, while rectal prolapse involves a higher segment of the bowel.

Causes
Rectal prolapse is caused by weakened muscles that hold the rectum in place.

Treatment
Hemorrhoids can be treated to help patients feel more comfortable and avoid complications. Rectal prolapse usually requires surgery to fix.

Progression
Rectal prolapse tends to develop gradually, starting with the prolapse coming out after a bowel movement and then returning to its normal position.

Because neither condition has specific symptoms, it's important to see a medical specialist for a diagnosis. A doctor may perform a physical exam and request additional tests, such as an x-ray or anorectal manometry.