Privacy settings - hide maps only. Is it possible?

Yes!!

This is a must. I'd like to share the Garmin Data publicly on my site, but not with my home as the start and stop. I know I could start and stop it away from my home, but why not just include a way to not share the private part of the data?

Is there any other way to share my workouts without exposing my home address?

Yes, by sharing screenshots containing only the information you want to share, without allowing the viewer any user interaction with the data.

As Dill said, screenshot is best.

Or do what I do. Walk a couple hundref meters away from home and then start gps.

I know I could start and stop it away from my home, but why not just include a way to not share the private part of the data?

Different individuals may consider different piece of information to be private or sensitive.

Never mind the idea that you – as an example of one individual – may choose to start and end your tracked runs or rides from your home address, and your home address is private and/or personal information that may lend to your identification. Garmin has no concept of your home address in the context of activity data, and some users (such as Anton_Vaasa and myself) don't start or finish workouts right outside home, so there is no inherent logical equivalence of start location (or end location) of a workout being one's home address and therefore ‘inherits’ the same data sensitivity as one's home address.

Yet I, as another individual user, may consider information about my health to be private and sensitive (in line with privacy legislation in Australia), and the heart rate chart for a workout – especially when correlated with pace and cadence – would disclose significant indicators of my cardiovascular fitness and overall health.

Yet other users may consider their actual performance (e.g. best pace, average moving pace) to be private, but wish to share other aspects of activity data, such as the routes they ran, or heart rate as recorded by their watches because they are discussing issues to do with data accuracy of their watches' built-in optical heart rate monitors.

I think it is unreasonable and unrealistic to expect Garmin to allow every data class and/or attribute to be assigned a user-selectable privacy setting for the purpose of governing what is shared. No individual user's views or preferences ought to be considered representative or typical, and in fact the classification of data of particular information – such as home address or health information – may be different from nation to nation, or from one jurisdiction to another in accordance with the applicable privacy laws and regulations. The easiest and safest way to proceed is simply to allow users to voluntarily and expressly choose between sharing everything or sharing nothing, with no allowance for more fine-grained controls.

Use the privacy zones to hide certain areas. 

Also send the idea of map privacy on/off to Garmin: https://www.garmin.com/forms/ideas/

Smil3yGirl said:

I think, contrary to one of these comments, that it's perfectly reasonable to ask for a "map on/off" toggle.

I agree but it's par for the course for Garmin to be missing features that Strava (for example) has had for years. Honestly I don't know any runners who use the social aspects of Garmin Connect anyway. Everyone I know uses Strava for that stuff. I personally keep all my Garmin Connect activities private.

As far as that comment goes, ironically, Strava also allows you to hide HR, pace, calories, and other activity details.