Ok, after playing around a bit more, I can observe that this is a "normal" behaviour.
So after a library update is published / build in Launch, it seems like it requires to launch the Adobe Launch Library Download workflow twice on AEM before the libraryUrl correct itself.
The only thing pending is that the rule using the Core library to run Custom Code is not running on the page.
Can you please let us know how are you trying to run custom code?
I have added an alert message in this example  and it is working fine for me.
I tried running the following custom code:
document.title="Loaded from Launch"
Using the Core extension:
And the rule is just simply:
When I look at the source of the Launch script, I can see the Rule that I have setup above and it references another JS file with the following code:
_satellite.__registerScript('http://<AEM>:<PORT>/etc/clientlibs/dtm-reactor/<ENV-hash>/BLa1ad5f860f084972a62d7c090679951b/RC26 15a6f895d94e3e9e7d367a74d3c621-source.min.js', "document.title=\"Loaded from Launch\";");
However, the page is not executing the script as I cannot see the page title change.
For me, basic alert function did worked. I am not very sure about how Launch let JS work inside the web page.
Adobe Launch is to insert Tags into to web pages and retrieve the values the web-page. I am not very sure if using Launch for setting would be best practice. If you aim is to save some edited/appended value to the existing page title then that you can easily so by using Data Elements in Adobe Launch.
Jantzen.Belliston Adobe Launch expert need to intervene here to clear some more doubts here.
Also, for the initial issue:
The libraryUri path was not referencing the latest version (points to libraryEntriesPrev rather than libraryEntries) in AEM after triggering the workflow to download from Launch until a 2nd workflow was triggered.
- Is this the normal behaviour?