How can I make calls related to a case display in its Chatter feed?

I am using the Service Console in Lightning Experience, and using the Chatter Feed component on the Case layout to display updates to the Case.

If I use the "Log a Call" Quick Action from the Chatter publisher, these calls are displayed in the Chatter feed. This is the desired behaviour.

However, if I insert a Log a Call record related to the Case any other way - e.g. using the Developer Console, or my CTI system's custom components - the call does not display in the feed.

If I query the record, it looks exactly the same as the record that is created with the Quick Action. It just seems that the Chatter post is never made in this situation.

I have checked my Feed Tracking settings:

• Task has Enable Feed Tracking and Subject checked
• Case has All Related Objects checked

How can I enable calls created with a custom component (part of a CTI system) to display in the case feed?