Can we report on contributions by payment processor?
There are no such reports in CiviCRM that will filter Contribution by Payment Processor. Payment Processor and Payment Method have different meaning. Payment method can be used by N processor. Here are some solutions


Going forward have your system to use unique payment method for a payment processor with same name and then you can filter the report using Payment method. So that in future it can be tracked easily using payment method. However, for historical data I won't recommend you to update the Payment method since you will need to update in two tables i.e civicrm_contribution.payment_instrument_id and civicrm_financial_trxn.payment_instrument_id. Changes to civicrm_financial_trxn might impact on Bookkeeping entries for those which are exported in batch and closed for accounting period. 
Build a custom report for contribution having Payment processor as filter and/or column.
Use AlterReportVar hook to add filter and/or column for existing contribution reports and build query accordingly.


The basic query to retrieve contribution done using payment processor is 

SELECT cc.id FROM civicrm_contribution cc
  INNER JOIN civicrm_entity_financial_trxn ceft
    ON ceft.entity_id = cc.id
      AND ceft.entity_table = 'civicrm_contribution'
  INNER JOIN civicrm_financial_trxn cft 
    ON cft.id = ceft.financial_trxn_id
WHERE cft.payment_processor_id = 1;