Wow!
There is a key to all of these preperations and methods. That key is acid.
Nearly any acid can stimulate root growth in woody plants. Willows, long time stimulant for roots is because of salicyllic acid, that which aspirin is made of. The acid is named for the plant, willow = salix.
IBA, indole-3-butyric acid, Is a synthetic acid that was found to induce rooting in woody plants. This is found in many cloning powders.
IAA, indole-3-acetic acid, is found in potatoes, which also has been found to induce rooting in woody plants. In many countries potatoes are used to make vinegar, acetic acid, as well as the famous Wodka.
The onion contains many sulphur compounds while forming the bulb. When cutting an onion, the sulphur fumes mix with the water in the eyes which causes irratation and crying. The irratation is caused from the fumes making sulphuric acid in the eyes. When rubbing the wet cutting on the onion, one makes this acid which works like many acid auxins and stimulates roots.