It really depend upon what functionality you need from the map.
If you simply want to embed a map with a single marker on a particular place, as in the example you linked, that’s quite simple without needing the API.
(observe the source code of that site)
If you need something more complex, such as multiple map location markers, custom overlays and other custom map features, you will need to use the API.
I suggest that on mobile screens you don’t let the map go edge to edge otherwise the user can get trapped inside the scrolling map and not scroll the page unless there is a gap at the side.
You can get the embed code directly from Google Maps.
Click on a featured place to bring up the info in the side panel.
Click on Share.
In the Share box click “Embed a map”.
Copy the embed code and paste into your HTML page.
I have switched to using OpenStreetMap to avoid possibly having to pay fees for using Google maps:
And the graphics on OpenStreetMap is much better too!
When viewing a map with OpenStreetMap you can get HTML code to embed via the ‘Share’ button on the right hand side, with the option of including a marker.