A certificate has two [optional] extensions that direct how this certificate: Key Usage and Extended Key Usage. Prior to version 11.0.9 Acrobat did not process these extensions properly according to RFC 5280. Starting with version 11.0.9 Acrobat strictly follows the RFC 5280 restrictions on the certificates usage. MS Word does not follow these restrictions.
Go to Edit->Preferences->Signatures->Identities & Trusted Vertificates->More... Select Digital IDs in the right pane and in the left pane your certificate. Then click on "Details" and look (you may need to scroll), if it has an "Extended Usage Key" extension. If it does click on it and look at its value in the bottom part of the left pane. If it has emailProtection or CodeSigning value then this certificate is suitable for signing in Acrobat. If it does not have any of these values but has some other value, like clientAuthentication, then it is not suitable for signing in Acrobat. The problem in the past was that Acrobat allowed the users to sign with certificates that were issued for purposes other than document signing. Version 11.0.9 tightened this restriction.
Thank you for a clear and thorough answer. I wish MS Word would follow the RFC 5280 also, instead of letting users take the wrong path.