Unfortunately I don't think there is an API that lists custom activity types. In fact, I'm not 100% sure the engagements API even returns anything other than the default (note, call, meeting, etc). You can create custom call dispositions but that's not quite covering the use case here. 😞
Thanks for your support so far. You have mentioned how to pass activityType to engagement api but i also need an api to get all activityTypes. Please share this as well