RE: Only one of two Ethernet network adapter works on my Dell pe2950 II, running VMware esxi 6.0
Both two works fine on CentOS.
If that is the case then this is a driver issue. ESXi 6 is not a validated OS for the 2950. The required driver for the LOM to function properly is not likely present in the OS. If you want to make it work then you could try going to VMware's site and downloading older VIBs for the LOM that may work better.
I have heard of people running ESXi 5 on 2950s, so maybe you should check to see which network driver it uses and then download that VIB from VMware and install it.