What is the Buddhist definition of "Atta"?
From the viewpoint of Buddhist practise, as explained in SN 22.81, 'atta' ('self') refers to an ignorance created 'thought formation' (sankhara) that ignorantly assumes self-identity &amp; ownership in relation to any of the five aggregates. 

This ignorant assumption of self-identity &amp; ownership is central in the generation of suffering, as explained in SN 22.1 and as explained in Dependent Origination (SN 12.2).

Therefore, in Buddhism, the primary importance of 'atta' is it is a central cause of suffering that is ever-present in the mind of the unenlightened person ('uninstructed worldling'). 


  Here, bhikkhus, the uninstructed worldling ... assumes form as self (attato). That assumption, bhikkhus, is a formation (saṅkhāro). That
  formation — what is its source, what is its origin, from what is it born (jatiko)
  and produced? When the uninstructed worldling is contacted by a
  feeling born of ignorance-contact, craving arises: thence that
  formation (of 'self') is born. SN 22.81





  There is the case where an uninstructed, run-of-the-mill person — assumes form (the body) to be the self or the self as possessing
  form or form as in the self or the self as in form. He is seized
  with the idea that 'I am form' or 'Form is mine.' As he is seized with
  these ideas, his form changes &amp; alters, and he falls into sorrow,
  lamentation, pain, distress &amp; despair over its change &amp; alteration.
  
  There is the case where a well-instructed disciple of the noble ones — who has regard for noble ones, is well-versed &amp;
  disciplined in their Dhamma; who has regard for men of integrity, is
  well-versed &amp; disciplined in their Dhamma — does not assume form to be
  the self or the self as possessing form or form as in the self or
  the self as in form. He is not seized with the idea that 'I am form'
  or 'Form is mine.' As he is not seized with these ideas, his form
  changes &amp; alters, but he does not fall into sorrow, lamentation, pain,
  distress or despair over its change &amp; alteration.
  
  SN 22.1





  And what, bhikkhus, is attachment? There are these four kinds of attachment: attachment of sensual pleasures, attachment of views,
  attachment of rules and vows, attachment of words of
  'self' (attavā­du­pādā­naṃ). This is called attachment.
  
  And what, bhikkhus, is birth? The birth of the various beings (sattānaṃ) into the various orders of beings (sattanikāye)...
  
  SN 12.2


Note: 'Satta' is a synonym for 'atta', in 'satta' is also a view about 'self'. Refer to SN 5.10 &amp; SN 23.2. 




  Therefore, bhikkhus, any kind of form whatsoever ... Any kind of feeling whatsoever … Any kind of perception whatsoever … Any kind of
  volitional formations whatsoever … Any kind of consciousness
  whatsoever, whether past, future, or present, internal or external,
  gross or subtle, inferior or superior, far or near, all consciousness
  should be seen as it really is with correct wisdom thus: ‘This is not
  mine (mama), this I am (asmi) not, this is not my self (atta).’ SN
  22.59
  
  
  
  It is said that the world is empty, the world is empty, lord. In what respect is it said that the world is empty? Insofar as it is
  empty of a self (attena) or of anything pertaining to a self
  (attaniyena): Thus it is said, Ananda, that the world is empty Suñña
  Sutta