They both work for me in 2023.1.6 in the classic and the web UIs.
(On BAQDesigner.GetByID - which previously got no messages)
Plot thickens. I did a PublishInfoMessage for Ice.Common.BusinessObjectMessageType.Error,
and got all my messages.
I removed it, and still got all my messages.
Inconsistent too, not getting it on POForm.SubmitToAgent at all, no matter what I do.
Just to make sure that BPM was running in post, I decided to divide by 0.
It was very unhappy, so yes it’s firing 
Thank you all for taking your time to look at this.
Support is currently telling me it is related to this problem record:
PRB0266795 - Current status, In Progress
Data Form no longer works in Classic/Modern environment after 2023.1 Upgrade when using the Erp.Rpt.ReportName.SubmitToAgent Method
Slated for fix in 2023.1.10 
They sent me a test fix for PRB0266795.
All 3 message options worked, as well as the BPM Data form.
From the BO in question.
I will post here if they say anything more on release schedule.
Development has confirmed this will be in the next update 2023.1.10 and appreciates your testing. Please let us know if you need anything else in the future