Limits of SQLite

That page you linked, besides being quite old, talks about accesses from the same process through the same database connection (or through multipe connections in shared cache mode, which you should not use).

When not in WAL mode, multiple connections can read from the same database, but a writing transaction is exclusive, i.e., no other readers or writers are allowed.

In WAL mode, a writer and readers do not block each other, but there is still only one writer allowed.

your first query is standard SQL (the parentheses are redundant, only needed by MS-Access) and should work just fine in SQLite, with or without the parentheses:

SELECT 
    Message.messageContent, Message.messageDateTime 
FROM
        Message 
    INNER JOIN 
        ( Contact INNER JOIN contact_m2m_message 
            ON Contact.contactID = contact_m2m_message.contactID
        )
      ON Message.msgID = contact_m2m_message.messageID ;

Edit: Actually, SQLite needs the parentheses, too, at least the version provided in SQL-Fiddle.

The 2nd query is not valid because you have not provided an alias for the derived table:

SELECT 
    Message.messageContent, Message.messageDateTime 
FROM 
        Message 
    INNER JOIN 
        ( Select Contact.contactNumber, Contact.contactName 
          FROM Contact INNER JOIN contact_m2m_message 
               ON Contact.contactID = contact_m2m_message.contactID
        )     --<-------------------------------------- no alias
            ON Message.msgID = contact_m2m_message.messageID;

To make it work wirh a derived table, you have to provide an alias for the derived table and also add the contact_m2m_message.messageID in the SELECT list of the derived table (and remove columns that are not used):

SELECT 
    Message.messageContent, Message.messageDateTime 
FROM 
        Message 
    INNER JOIN 
        ( Select -- Contact.contactNumber, Contact.contactName,
                 contact_m2m_message.messageID 
          FROM Contact INNER JOIN contact_m2m_message 
               ON Contact.contactID = contact_m2m_message.contactID
        ) AS cm
            ON Message.msgID = cm.messageID;

But I don't think you should use that. The first query should work. You can re-arrange the order how of the tables are joined. Using table aliases also helps in readibility:

SELECT 
    m.messageContent, m.messageDateTime 
FROM 
    Message  AS m
  INNER JOIN
    contact_m2m_message  AS cm
      ON m.msgID = cm.messageID 
  INNER JOIN
    Contact  AS c 
      ON  c.contactID = cm.contactID ;