Does posture matter while drinking water?
The stomach is equipped with the pyloric valve to regulate transpyloric flow to the duodenum.
Both cold and warm drinks stimulate a pattern of motility associated with retardation of transpyloric flow.

The pyloric valve is responsible for acidifying water that you drink, and if the body's guard against amoeba's and dysentery depended on body position, it would be low survival performance, given that sedentary posture is associated with poorer health outcomes, including an increased risk of type 2 diabetes. You will benefit from minimizing time spent sitting each day, and from breaking up periods of time spent being sedentary, as often as possible. It's aperture at rest is 3mm, so it's of minimal difficulty for the stomach to hold water as required for optimal health. It's highly likely that water is kept for enough time to expose it to stomach acids, regardless of body position, and then it's transmitted.
Water doesn't flow straight through the stomach like a straight tube when it's empty. 
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1382811/pdf/gut00528-0043.pdf