As far as I'm aware the Display Template is configured on the actual search landing page. If you reconfigure the web part on the peopleresults.aspx page to use the custom display template it should render as you desire.
If you want to be able to render using both the default people display template and the custom one you created, then you simply need to result pages.
Update:
To configure a Result Type so that the changes will also be reflected on the regular Results.aspx page:
- Navigate to the Site Settings Page
- Under the Search category, select the Result Types link
- Select the New Result Type option
- Give it a name
- For the source, select Local People Results
- Select your Display Template
- Click the save button