The app action buttons on the "Record Details" screen are displayed in the order they are listed on the "Actions" setting screen (in the order from top to bottom).
To change the display order of app action buttons, first navigate to the "Actions" settings screen by referring to the following article.
Configuring App Actions

Then, click and hold the Reorder icon to the left of the app action you want to move and drag the app action to the position of your choice.

After you reorder the app actions, click App Settings in the message displayed at the top of the screen or click Back to App Settings displayed at the bottom left of the screen.

Click Update App on the upper right side of the screen, and click Update App on the confirmation dialog.
The new display order is not applied to the "Record Details" screen until you update the app.