Is bright red blood "woven into" stool more correlated with certain problems than blood that pools around stool?"

Maybe / Maybe not: Bleeding per rectum may originate in anal canal, rectum, or further up the line. Frequent anal sources of bleeding include hemorrhoids & fissures, that may respond to modifying stool texture (fiber, softeners) & topical agents, or require further intervention. Similar bleeds can be seen with anal & rectal cancer, infections, ischemic colitis, diverticular bleeds, inflammatory bowel, etc. See doc.