We have this issue coming from the huge Update Point button in the center of the page. I keep asking for them to change this or be able to turn it off but no. Crazy design choice really. At least call it Update location or something. For us we have 2 layers - one field and one main so we make 2 maps and then can turn off all editing for one of them in the web map. It would be really nice if we could turn off moves by layer or in the map but this is a service wide setting for some odd reason. For now what we do is disable geometry editing. To move a point we append in a new one (or copy it) and delete the old one. Not great but it may help you.
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