Grounded thermocouple vs "grounded" thermocouple
Omega's guide is focusing more on the thermocouple itself.  Analog Devices' article focuses more on the sugnal conditioning for thermocouples.   I'll explain what is implied by Grounded and Ungrounded in the Omega's guide.  In the context of the Omega's guide, the thermocouples are a part of thermocouple assemblies or thermocouple probes.  


Ungrounded thermocouple.  Thermocouple wires are not connected to the sheath of the thermocouple probe.


Grounded thermocouple.  Thermocouple is deliberately connected to the sheath to improve the response time.

Here's an example of how a thermocouple probe can be used.  It measures temperature of water in a tank.  The tank and water are grounded.  As a result, the sheath of the probe is grounded.  If the thermocouple itself is connected to the probe sheath, then the thermocouple is grounded to the tank, no matter how you connect the thermocouple wires.  If thermocouple wires are not connected to the sheath, you can connect the output ends as you see fit.