Chapter 2:

Salesforce Formula:

IF(
  RecordType.Name = 'Sales' && ISPICKVAL(StageName, 'Closed Won'),
  'NEW',
  IF(
    RecordType.Name = 'Renewal' && ISPICKVAL(StageName, 'Closed Lost'),
    'CHURN',
    IF(
      RecordType.Name = 'Renewal' && ISPICKVAL(StageName, 'Closed Won'),
      IF(
        ARR__c - ARR_Up_For_Renewal__c > 0,
        'UP',
        IF(
          ARR__c - ARR_Up_For_Renewal__c < 0,
          'DOWN',
          'FLAT'
        )
      ),
      NULL
    )
  )
)

Explanation:

Chapter 3:

Salesforce Formula:

IF(
  RecordType.Name = 'Sales' && ISPICKVAL(StageName, 'Closed Won'),
  'NEW',
  IF(
    RecordType.Name = 'Upsell' && ISPICKVAL(StageName, 'Closed Won'),
    'UP(amd)' ,
    IF(
      RecordType.Name = 'Renewal' && ISPICKVAL(StageName, 'Closed Lost'),
      'CHURN',
      IF(
        RecordType.Name = 'Renewal' && ISPICKVAL(StageName, 'Closed Won'),
        IF(
          ROUND(ARR__c, 2) - ROUND(ARR_Up_For_Renewal__c, 2) > 0,
          'UP(inc)',
          IF(
            ROUND(ARR__c, 2) - ROUND(ARR_Up_For_Renewal__c, 2) < 0,
            'DOWN',
            'FLAT'
          )
        ),
        NULL
      )
    )
  )
)

Explanation: