This takes about ten minutes, once. You'll make three automations in the Shortcuts app: one for leaving, one for arriving, and one for joining the home Wi-Fi.

Start

You'll need these two things

Keep the key private. It only works for your phone, and making a new one turns this one off.

First, one iPhone setting

1. Open Settings, then Privacy & Security, then Location Services.
2. Scroll to the bottom and tap System Services.
3. Turn on Alerts & Shortcuts Automations. Without it, your phone never tells the house anything.

When you leave

1. Open the Shortcuts app and tap Automation at the bottom.
2. Tap + (or New Automation), then choose Leave.
3. Tap Location, pick your home address, and tap Done.
4. Choose Run Immediately and turn on Notify When Run, then tap Next.
5. Tap New Blank Automation, then Add Action, and search for Get Contents of URL.
6. Tap the blue URL and paste the web address.
7. Tap the › arrow to show more. Set Method to POST.
8. Under Request Body, choose JSON. Tap Add new field, choose Text. Key: event. Text: leave.
9. Tap Add new field again, choose Text. Key: key. Text: paste your phone key.
10. Tap Done.

When you arrive

1. Make another automation the same way, but choose Arrive in step 2.
2. In the event field, type arrive instead of leave.

When your phone joins the home Wi-Fi

1. In Automation, tap +, scroll down and choose Wi-Fi.
2. Tap Network, pick the house's Wi-Fi, and tap Done. Leave it on Is Connected.
3. Choose Run Immediately. Leave Notify When Run off: your phone rejoins the Wi-Fi often, and the notices would get tiresome.
4. Tap Next, then build the same Get Contents of URL action as before.
5. In the event field, type wifi.

If your house has more than one Wi-Fi name (for example a separate 5 GHz one), you can pick all of them in step 2.

To check it: the next time you leave or come home, your phone shows a small notice, and your line under "Who's home" on the main page changes. Coming home, the house waits about five minutes before switching to Home, unless your phone joins the Wi-Fi first, so driving past doesn't count.