How trigger update in replica database on Postgres?
The problem with updates in your scenario is that my_table on the standby server does not contain the row being updated, so that the UPDATE does nothing and no row level trigger is fired. Since nothing was inserted into my_table, an UPDATE cannot be replicated.
If you want to replicate from a non-partitioned to a partitioned table with logical replication, use PostgreSQL v13 or better, where that is supported. You will bot be able to automatically generate partitions that way, but it shouldn't be a problem to create the partitions before you start replicating.