Hi
An Inbox Rule can be a client-side or a server-side rule. A server-side rule can be performed without any requirement from the local machine, for example: change the importance of a message; move the incoming message to a specific folder on the User Mailbox; delete a message and so on. The general rule is that all the components must be on the server side.
A client-side rule requires processing from the Outlook client, an example of a client-side rule is move specific messages identified by the rule to a PST folder. This rule cannot be processed at the server-level because the server is not able to work with a PST file on that context.
So From OWA, it should be a client side rule.
For more information
Cheers
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Please go to your Exchange Server and run the command
Get-CASMailbox "mailbox" | fl OWASignaturesEnabled
If it is disabled, please enable it.