The ECM power control is slightly different for the Classic vs the Touring and the Roadster
On the later models, the ECM power is supplied via the EMS relay - and in this case it is enabled by the ECM itself, when power is present on the A19 terminal (comes from the 'ignition' circuit)
When the 'ignition' circuit is turned off (whether by the key or kill/run switch) it starts an internal ECM timer, which does not immediately turn off the EMS relay, but provides for a short delay while the current trims are committed to memory (you will probably be able to hear the EMS Relay click off after a few seconds of turning off the ignition)
On the Classic, the EMS relay is not controlled by the ECM - directly from ignition power only.
Therefor that relay will turn off as soon as the Key (or Kill/Run) is turned off. And therefor so will power to the ECM.
There is no direct 'ignition' connection to ECM so there is no external trigger (that I can see) for a countdown after key-off on those models
(I may have to study the early circuit a bit more - might be missing something, but I can't see it at the moment)