Concerning the role of hypothalamus and the posterior pituitary gland in osmoregulation and especially when water potential of the blood is low, When the osmoreceptors in the hypothalamus detect the low water potential of the blood, they lose water by osmosis and shrink, my question is: NUMBER 1 》does this shrinkage stimulates the syntheisis of the ADH in the neurosecretory cells in hypothalamus first and then ADH is sent along the axons where they terminate in the posterior pituitary gland secondly and released into the blood stream thirdly (just non stop) , OR NUMBER 2 》 before the shrinkage happens, already ADH was synthesized in the neurosecretory cells and was sent along the axons where they terminate in the posterior pituitary and ADH was stored there in the axon terminals and whenever the neuroscretory cells shrink, impulses are sent along the axons stimulating the release of ADH from the posterior pituitary gland into the blood stream? Because I have read both explanations from different resources and I am confused which is the correct physiology mechanism that releases ADH from posterior pituitary gland.
Your second hypothesis is correct.
Hypothalamus secretes vasopressin (ADH), transports it along the axons and stores it in the posterior pituitary, thereby when hypothalamus senses the stimuli(osmotic potential or H+ levels or some other indicators) it generates nerve impulse and sends along the posterior pituitary to release that stored ADH.
Comparison of Both Cases
To get insight into the answer lets consider both hypothesis one by one and compare them on the basis of speed of action, in the first case as you mentioned if hypothalamus starts to transport ADH on stimuli this will lead to slow effect (and a comparatively long time / lag to show effect) also once if hypothalamus feels it needs to secrete more ADH since it wasn't enough to meet the requirements it will take even longer time which could be a disadvantage to the mechanism overall. Whereas in second hypothesis these limitations are take care of - hypothalamus transports it to Posterior pituitary from where it can deploy the effect quicker than earlier situation, also it can store some extra ADH in it's own once it has bulked the posterior pituitary's potential which is useful as a secondary backup.