Add default Application Name to connections

Currently we don't specify an Application Name, which means that unless the user specifies one themselves it will just show up as something generic like this :

`Core Microsoft SqlClient Data Provider`

which isn't very useful.

Users can always specify it themselves in their connection string, but the downside of this is that there's no easy way to tell which connection is for what purpose (and we make a number of connections, especially for the trigger).

So the first question is whether we should do anything about this at all - or just let people specify an Application Name themselves if they want.

Some other options if we do want to add it :

1. Assign a name ourselves to each connection, overwriting whatever the user may have specified
2. Check if the user already passed in a name - and if so then do nothing (leave it as-is)
3. Check if the user already passed in a name - and if so then use that name but append a specifier to the end to signify what the connection is doing. If none provided have a default value for each connection.

No files, tests, or entry points are named. First inspect how the extension creates its multiple SQL connections and handles connection-string Application Name values; clarify the desired default and override behavior before identifying tests that define done.

Written by the indexing model from the issue text.