Hi Daniel,
as my colleagues already said:
"<DIR_PUT>/abap/tmp" is the first directory. It contains the log during the writing process. If the upgrade phase finished then they will be moved to "<DIR_PUT>/abap/log" directory.
You can sort the logs by date/time. The newest one are that one from the current/last upgrade phase and contain the newest information.
Regarding your question with the upgrade phases - the modules are Execution (TABIM_UPG) and Preprocessing (SHADOW_IMPORT_INC & PARMVNT_SHD).
Best regards,
Andreas