I had the exact same symptoms as you have divulged here which started over 1 and a half years ago. I had stool tests to which one sample came back with a trace of blood. My GP wasn't too concerned but booked a colonoscopy anyway to which I was keen to do so we could get to the bottom of the symptoms. What came back was I had mild proctitis which is the inflammation of the rectum. The rest of my digestive tract and bowel was clear and fine.
My stools were coming out with some odd shapes (squared edges and lined indents) but in relation to the leaky coloured stools, I was out in the bush a couple of months ago on a trek and nature called whilst out. I had to go in the bush to which I examined my stools without it having been covered by the water in the bowl. I noticed it had orange mucus on the outside of the stool which is a symptom from the proctitus. This was why water would colour whilst being on the toilet for around the 10 minute mark. We have not found the cause of the proctitus but I am working on some remedies with diet etc. Hope this throws some light as to a possibility for others with the same symptoms.
There is nothing to be embarrassed about. We all poop! We all have questions at one time or another. I had, I kid you not, very green stool last week. You know what it was? Lucky Charms cereal, a lot of it, which I never eat but was stress eating and got into some summer (spoiling and not doing our usual thing) cereal for my kids. The die in the marshmallows, I've learned, is well known for changing stool green.
I would have gone to some kind of dye in your food for sure as the first thing to consider. Something else is supplements. Do you take any? If you take one that contains beta-carotene and aluminum hydroxide, that can change stool color to orange red. There is another reason which is medical, which is lack of bile salts. Lack of bile salts happens usually either because a person has the underlying conditions of Chrohn's diesease or Irritable Bowel Syndrome. Obviously, the other concern is that it is actually blood from your GI tract.
So, is there any way you can see the doctor? I would not think anything serious at this point but still think you should make sure it's not. :>)