That is strange behavior indeed. When I removed the extent portion of the url the points did not disappear.
As this is the AGOL WAB version you should call esri tech support and report this issue.
I grabbed your REST service and added it as an item and built a quick basic viewer. I did not experience the same phenomenon. How did you consume your ArcGIS Server service into AGOL (individual layer per item or entire service as an item)? I watched the communication via Fidler on your app and it was making hundreds of calls back to your ArcGIS Server. I consume3d your entire service and my app made significantly fewer calls. If you have Chrome, add the Fiddler Add-In and watch the communication. Might be the way you consumed your layers into AGOL or how they were added (maybe altered) in the web map.
Thanks, I added the layers individually. If I use the whole map service then the user would have to chick on the map service name to see all of the layers. Trying to make it easier for the user. No familiar with Fidler.
Okay, I suspect it has something to do with that....hopefully tech support can explain what is happening. Fiddler is a nice application to watch internet traffic. I included a snapshot below, but I could essentially see hundreds of calls to your AGS box when I started the application.
Be aware that with modern browsers like Chrome and Firefox fiddler is not really needed anymore. The network tab and developer tool features of modern browsers can do most of what Fiddler was designed for.