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Hello Georgi,
Thanks for your updates in PM with patience.
Based on the powershell result you provided in PM, the connector 3 you created from Office 365 to your partner organization is validated. However, from the both message trace results ( successful and failed), the successful one is indicated with connector 5, and the powershell command you performed was associated with connector 3, may I double confirm if you created different connectors from Office 365 to your own server for mail flow? You can go to Office 365 Exchange Admin Center > Mail flow > Connectors to double check it, if you have different connectors, please temporary disable/remove the connector 3 and just keep the connector 5 to see if it make any difference.
Moreover, since the connector you created is validated, however it seems not triaggered when you sent emails to your own server, may I know when you created the connector, which option you choosed when to use the connector " Like For email messages sent to all accepted domain in your organization or only when email message are sent to these domains"? Please double check the connector there, and if convenient, please choose the option " For email message sent to all accepted domains in your organization" at this time to see if it make any difference, thanks.
On another hand, I'd like to collect some more information for the accpted domains in your organization for further troubleshooting, please connect to the Exchange Online Powershell and then run the command Get-AcceptedDomain | FLafter that please share the full result to me in PM, thanks.
Best Regards,
Oliver