I've looked everywhere but can't find any documentation on how create an "Install App" button like this on http://storetasker.com/ (not my site, don't recommend them, just an example)
Sorry for the short question, I usually refuse to post on here without what showing what I've already tried but I can't even find where to start since the magic that happens on the aboves /merchant-install
The app is unlisted; that is - shopify won't host it
You can build the url from the instructions described on Shopify's authentication documentation.
https://help.shopify.com/en/api/getting-started/authentication/oauth#step-2-ask-for-permission
One I made for myself looks like this:
https://my-store.myshopify.com/admin/oauth/request_grant?client_id={api_key}&redirect_uri=https%3A%2F%2Fmy-app.com%2Fauth&scope=read_themes%2Cwrite_themes%2Cread_products%2Cwrite_products%2Cread_customers%2Cwrite_customers%2Cread_orders%2Cwrite_orders%2Cread_script_tags%2Cwrite_script_tags&state=5c9f319dcc43d72fe7
Please refer to API Access Scopes for a list of possible scopes.