How to track automatically where lat, lon are sent to my personal email address with inreach?

You can manually send lat/long coordinates to a private email address, not automated, and...

You can automate sending Tracking to Mapshare, which shows up as dots, not coordinates. You can make your Mapshare page private.

But you can't do what you are suggesting.

thx Like2Hike if you're right, then inReach for me is not the way to go, and I might have to use the Iridium Go! then. That does support this feature. 

Thing with most tracking providers seems that they all like to collect the data themselves, to publish it on their platform. I'd prefer to publish it on my own website, on my own map...

THIS part (publishing the points on your own web page) is possible via a KML feed. You might be able to use the KML feed to send emails. There are also more powerful APIs. However, the APIs are only available on professional accounts.

Summary information here:

https://support.garmin.com/en-US/?faq=Oa5mP2D5Zf7NZ8P17ATN58&productID=623975&searchQuery=api&tab=topics

This one, which is linked from the above, gives further info on the KML feed.

https://support.garmin.com/en-US/?faq=tdlDCyo1fJ5UxjUbA9rMY8&productID=623975&searchQuery=api&tab=topics

If you are going to publish on your own website/map, that is different than a personal email address. You can use the KML feed, if your website/map is compatible. The feed is found within the Explore website, under the Social tab. 

It's additional. I want to email them to my website and then (also) publish them on the website. I could retrieve an KML file and read that, though ... only if that is officially supported and won't change over time when I am sailing of course.

Thx a lot. I'll dive into that.