M109A6

Looks like the easiest way is to select Tools > Data > Batch Editor then choose OT_Name, select Add, then put in the name. Example: .OT_Name=NewTrainer Select run and it should update the name for all Pokemon in Boxes or Party, whichever tab is chosen. If you want to do both then you'll have to have Boxes selected and choose Run, then select Part and click Run again. As always, be sure to make a backup before you make any changes.

Changing the name in the SAV field changes the trainer name. When you catch a Pokemon going forward it will have the trainer name you chose. Changing it in the OT/Misc section is Pokemon specific to a Pokemon being edited. For example: After you change the name in SAV, right click a Pokemon and choose ‘View’, then update the trainer name, and right click the Pokemon you just chose view on and choose ‘set’. This will override the Pokemon in that slot with the updated trainer name.