That’s correct AITpro. A valid SSL certificate needs a trusted root CA for it clear warnings and have that security level. However, the heart bleed bug affected all unpatched OpenSSL 1.0.1 and the any certificates created and served by that software.
Also, most trusted root CA that you buy SSL’s from will reissue your SSL free of charge. Just contact them and ask on how to process an SSL reissue.
In regards to the whether the host will do everything expected, well, that is really a question of trusting your host and service provider. I can safely say that any serious host or online service that was affected would patch the OpenSSL software, revoke all certificates, reissue the certificates and reset all critical login credential passwords or at least inform the account holders that they should update their passwords to mitigate the possibility that a hacker gained those credentials via the heart bleed exploit. However, verifying that directly with your host or service provider is best practice.