I attach a little stack which you can use to test if you get this "system dialog from hell" on your Mac - try dragging the red square.
A first I thought the difference lay in the fact that the LC IDE has in its info.plist an entry for
which is missing from the OxTLite info.plist. The entry really should be there (it is in the OxT DPE plist) but it actually makes no effective difference, I think the entry merely supplies a reason to the user as to why the app is making the request.<key>NSScreenRecordingUsageDescription</key>
A Google search finally pointed me to the solution to making the permission "stick". First I deleted the entries in the System Screen Recording Authorised list for all 4 apps LiveCode, OxtLite, OxT DPE and my standalone. Next I launched each app in turn and attempted the stack drag operation: in every case the os did ask for permission as expected on the first drag. So Add each app to the system list. Now turn OFF permission, and Quit the app ( the Google correspondent recommended Force Quit). Now turn permission back ON, and relaunch the app. In each case I no longer get the permission dialog even after relaunching the app or rebooting the Mac, so the permission is sticking - so far anyway. It is evident that it is a further Ventura, or maybe earlier, os bug. I don't know if it is fixed in Sonoma because I haven't dared to install Sonoma in case OxTLite breaks (can someone reassure me on this because the forum posts have me confused as to whether or not is now OK to use Sonoma? and the next os?)
In the course of this investigation, I came across two minor issues...
1. Every time I launch the latest OxTLite I get a dialog about "Refactor" now being a feature. Thank you, got the message, now could it please go away, never to be seen again?
2. When I actually add OxTLite to the Screen Recording Permissions list, it appears as "LiveCode 10.0.0 (dp)" !!! Whereas OxT DPE appears correctly as "OpenXTalk 1.963 1rc3". My guess is there is an incorrect legacy entry in the info.plist, possibly
<key>CFBundleSignature</key>
<string>Revo</string>