This is openHAB. There is almost never “the only way.” There are always multiple ways.
The API is at least somewhat public, it’s not clear if that SDK comes from GE or it’s a reverse engineered effort. You could figure out how to use that external SDK (or the Python one Michiel posted) to interact with GE’s cloud services. From there you can send and receive messages with OH through the REST API or MQTT.
You could go a little deeper into the API and implement it as OH rules using the HTTP binding and/or sendHttpXRequest Actions. Usually the hardest part is if the API uses OAuth2 but I’ve shown how to do that with rules at OAuth2 using just OH Rules and myopenhab.org. I’ve been experimenting off and on trying to get the Honeywell Home API working in this way but I can’t get past the request for the auth token even when using CURL, which is frustrating.
You could put a smart plug on the washer that reports power usage and Washing Machine State Machine to determine when the load is done.
You could use other indirect sensing methods such as vibration, noise, temperature of the hot water pipe leading into the washer, etc.
Don’t get too focused on how you think the problem needs to be solved (i.e. get the notification from the phone app into openHAB) to the exclusion of a whole host of other approaches that solve the actual problem just as well or better and perhaps with less work.
They used to have a Maker channel (though I think it was renamed) that would make outgoing webhook calls. Do they not any more? I deleted my account a long time ago.
Unless there is a plugin I don’t know about, which is always a possibility, there is no way to directly trigger Tasker via an email. Obviously, the notification from the email app can be used though. And there is at least one MQTT add-on so that email-mqtt could be used. I think the Android app has/is getting support to trigger a profile based on commands to Items maybe so openHAB itself could command Tasker to do something based on an email.
It is probably worth your while to spend that effort exploring the two API implementations already linked to here and writing a script that lives on your OH server to bridge between the two. Using a spare phone and Tasker in this way seems like just as much work but the end result will be flakier and brittle.
You don’t need IFTTT nor Tasker for that. Set up GA integration with OH and your “I’m cooking” routine can send a command to an OH Item and do it’s thing from there in a rule.
Honestly, this is how it goes for all technologies and APIs that OH supports. To be successful you must be familiar with the underlying technology. Sometimes that means a lot of research and learning and experimentation, sometimes not so much. But it’s rarely pressing the big read easy button and it all works.