How do I make data from OpenStreetMap appear on a Garmin device?

Garmin maps, like rendered web maps, do not display every feature that exist in the data. Also, note that gramin maps available for download may not have the most up to date data. Some may be a day old, some much longer. Check where you downloaded the map.

AFAICS, the default mkgmap style does not render man_made=petroleum_well, so unless you downloaded a specialised garmin map, it's unlikely that they include wells. Where did you download the map ? You could try asking the garmin map author to add wells.

Or you could create the garmin map yourself. Once you managed to create a standard map, edit the default style to add wells. I haven't tested it, but adding this line to the examples/styles/default/points file should work:

man_made=petroleum_well [0x6416 resolution 22]

See the general mkgmap style documentation and the list of POI types for details.

If the wells are not rendered by the Garmin map maker, you can do two things:

1) Submit a request to render those features

2) Use http://overpass-turbo.eu and make a gpx file to use it in your Garmin