If I understand correctly, you are running a SQL Server instance on your local home network and you cannot connect remotely from the internet.
To verify network connectivity, run the command below from a command-prompt on the remote machine, replacing "127.0.0.1" with the actual IP public address you are trying to connect to.
powershell -Command "tnc -ComputerName 127.0.0.1 -Port 1433"
If the above fails but www.canyouseeme.org succeeds, it suggests your public IP and/or port 1433 is not allowed from the remote machine.
Note that the SQL Server Browser and UDP are not involved when you specify only the IP address or host name in the connection string. Only port TCP 1433 is used in this scenario.