There is a catch in case of cases and work orders objects when it comes to quick actions. In the article under Actions on record pages there is note that says
Actions on cases and work orders appear in a different way than on
other records. On case and work order records, the page-level action
menu contains custom buttons and supported standard buttons. Quick
actions appear on the Feed tab.
For other objects, you can edit page in the Lightning Experience and then select Salesforce1 and Lightning Experience Actions from the selection panel. The panel states
Actions in this section are predefined by Salesforce. You can override
the predefined actions to set a customized list of actions on
Salesforce1 and Lightning Experience pages that use this layout....
Once you click override the predefined actions
link, it will give you available actions for SF1 and lightning to use. And then position your lightning-action in your desired sequence, as shown below.
I worked on the Account object for this example, it should be no exception for other objects, and this is what I got
It should be noted that you can't add more than three (3) viewable lightning-actions on the page, as shown in the picture above in green square You can display upto 10 actions on page layout, but other action will available when you click the drop-down next to the last button on the right. There is an idea on this to let users display more Action Buttons on Lightning Highlights Panel, you can upvote that if it affects you.
Edit
I have just come across this Page Layouts in Lightning Experience trailhead and it says
The highlights panel, which contains key fields for the record, is the only part of a record page that you can’t customize using the page layout editor. The fields in the highlights panel are customized using a compact layout.
I just edited the highlights panel using page layout editor, strange. May be this trailhead content is old, if someone can put some light on this.
Adding another answer that may help people that stumble across this in the same situation I did - I just spent a fair amount of time spinning my wheels searching for a solution, and didn't think to include "Case" in my search.
If the highlight panel you're attempting to add actions to is for Case, note that Quick Actions added to the Case highlight panel actually appear on the Feed
tab on the layout.
Actions on cases and work orders appear in a different way than on
other records. On case and work order records, the page-level action
menu contains custom buttons and supported standard buttons. Quick
actions appear on the Feed tab.
Source: https://help.salesforce.com/articleView?id=actions_in_lex.htm&type=5
Best Answer
Actions on user profiles, cases, and work orders can appear in a different way than on other records.
Actions on the user profile page come from the Quick Actions in the Salesforce Classic Publisher section of the global publisher layout. Only standard Chatter actions appear on the user profile page, regardless of which actions are assigned to the User Page Layout or the global publisher layout.
When feed tracking is enabled for cases or work orders, the page-level action menu on those records contains only custom buttons and supported standard buttons. Quick actions appear on the Chatter tab.
Actions in lighting experience