The problem was solved in both cases, by re-importing the site's SSL certificate after it'd been modified to specify the 'Microsoft Enhanced RSA and AES Cryptographic Provider' as the CSP.
This can be done using the OpenSLL command,
openssl pkcs12 –export –in cert.pem –inkey key.pem –CSP "Microsoft Enhanced RSA and AES Cryptographic Provider" –out certAndKey.pfx
It can also be done using the certutil command.
For more details please refer to this article.