Type:     PDOException
OW Debug - Warning
Message:     file_put_contents(/home4/xxx/public_html/website/users/ow_log/error.log): failed to open stream: No such file or directory
File:     /home4/xxx/public_html/website/users/ow_system_plugins/base/classes/file_log_writer.php
Line:     61
OW Debug - Warning
Message:     chmod(): No such file or directory
File:     /home4/xxx/public_html/website/users/ow_system_plugins/base/classes/file_log_writer.php
Line:     65

that means you need to put 777 permissions for the ow_log folder

this error:

OW Debug - Exception
Message:     SQLSTATE[42S22]: Column not found: 1054 Unknown column 'context' in 'field list'
File:     /home4/xxx/public_html/website/users/ow_core/database.php
Line:     707
Trace:     

#0 /home4/xxx/public_html/website/users/ow_core/database.php(707): PDOStatement->execute()
#1 /home4/xxx/public_html/website/users/ow_core/database.php(435): OW_Database->execute('UPDATE LOW_PRIO...', Array)
#2 /home4/xxx/public_html/website/users/ow_core/database.php(475): OW_Database->query('UPDATE LOW_PRIO...', Array)
#3 /home4/xxx/public_html/website/users/ow_core/database.php(547): OW_Database->update('UPDATE LOW_PRIO...', Array)
#4 /home4/xxx/public_html/website/users/ow_core/base_dao.php(233): OW_Database->updateObject('ox_base_user_on...', Object(BOL_UserOnline), 'id', true)
#5 /home4/xxx/public_html/website/users/ow_system_plugins/base/bol/user_service.php(699): OW_BaseDao->saveDelayed(Object(BOL_UserOnline))
#6 /home4/xxx/public_html/website/users/ow_core/application.php(225): BOL_UserService->updateActivityStamp(1, 1)
#7 /home4/xxx/public_html/website/users/index.php(51): OW_Application->init()
#8 {main}

means that the update has not been successfully completed. You need to restore your back and follow the instructions from this topic again: http://www.oxwall.org/forum/topic/24133

as to this error: 

OW Debug - Notice
Message:     Undefined offset: 1
File:     /home4/xxx/public_html/website/users/ow_system_plugins/base/bol/theme_service.php
Line:     607

first of all this is a notice and you can neglect it. But to make sure it goes away, check whether you have not enabled dev mode and debug mode at the same time. 

Yes, these are native Oxwall plugins. 

Never keep debug mode and dev mode enabled at the same time. Only one at a time. Probably which is why you had unsuccessful update. 

While doing update keep only debug mode to true, profiler and dev mode should be false.