How does HTTPS certificate presentation work, exactly?
I'd agree with your guess that this can be caused by different certificates being used on different systems, which can explain the inconsistent behavior; depending on whatever affinity rules the vendor uses to distribute requests, this can also explain why someone (such as DigiCert's testing tool) gets consistently wrong results.
However, trying to guess what's happening is completely useless here; you should report this issue to the vendor's support and ask them to fix it.