How can I get the last inserted id from an specific table from a given stored procedure?
From here: (My emphasis below)


  The ID that was generated is maintained in the server on a
  per-connection basis. This means that the value returned by the
  function to a given client is the first AUTO_INCREMENT value generated
  for most recent statement affecting an AUTO_INCREMENT column by that
  client. 
  
  This value cannot be affected by other clients, even if they
  generate AUTO_INCREMENT values of their own. This behavior ensures
  that each client can retrieve its own ID without concern for the
  activity of other clients, and without the need for locks or
  transactions.


A fundamental point about LAST_INSERT_ID() is that bit about per_connection - if it wasn't on that basis, then the function would be useless in a multi-user environment. 

Unless there are further complications, you have absolutely no need to resort to the MAX() function - and you have no guarantee anyway that (in a multi-user scenario), your MAX(xxx) value won't be incremented by another user in the instant after you call it.