Which of these two MYSQL queries is better (and what is the default join type)?
The default join type is inner join.  

The second query may be better as it avoids an extra join to the partner table.  The required partner id value is available from the promotion table.  The impact will likely be minimal in this case. 

I would recommend prefixing the primary key id columns with the table name (singular form); play_id, promotion_id and partner_id.  Most design tools allow this, but may default to the form you have.

Learn to use explain plans to understand how the query will be executed.  This is not foolproof as the execution plan may be influenced by table sizes and other factors.  Different databases may generate different execution plans.