I suppose that "пригляди за ним" (also: присмотри за ним) is correct term and implies very focused attention. Like supervising continuously without taking your eyes away for a moment. Like indeed babysitting, cause it might take just few minutes for a baby to crawl and fall out the window or something. Because of that, it usually means relatively short timespan.
"Погляди за ним" I would expect to hear in some villages, where language is less literature and more dialectical or even archaic. It would imply more state of affairs than specific action. Something that goes for long but does not have to be continuous. Say, if one babushka goes for few weeks into town hospital, she may ask her neighbors to look after her hens. Which in practice would mean opening their shed in the dawn and closing it in the dusk and giving them meal and water thrice or twice a day. And in between they are walking by their own. Because that activity is mundane and does not need a lot of time and effort invested it may continue for much longer than for example supervising a baby.
W.r.t. "глядеть" it can be used but mostly in different verb forms and with different meaning.
You can not correctly order "гляди за ним" (to me it sounds very village-speak) I believe, but you can order "смотри за ним". Additionally, you can order "гляди в оба" ("watch with your both [eyes]", meaning "stay alert").
Then you can ask
- Ты не глядишь за ним?
- Ты глядишь за ним?
- Ты не присматриваешь за ним?
- Ты присматриваешь за ним?
Those questions query the state of affairs. Are you supervising someone/something or not. When your examples were requests politely worded as a questions ( "would you not follow me?" reportedly say UK policemen when arresting you ) those are just questions.
Well, there can be "Ты разве не глядишь за ним?"/"Pазве ты не глядишь за ним?" = "Aren't you watching after him?" where the emphasized surprise implies deviation from the normal, expected situation. This borders with "polite request worded as question" case, but still is not quite that.