Thermocouple Type-K: 3 Wires & Ceramic Shield?

[Asymptotic]

My guess the third wire is to help with noise imunity while preventing the whole thermocouple from floating. I would think there would be better noise immunity when this third wire is grounded back at the amplifier/transmitter than just grounding it at the weld. I would think of it as a shielded twisted pair microphone cable.

A propensity for noise issues depends on controller instrumentation amp design as well as TC construction and wiring. I ran into this when replacing an ancient Barber-Coleman MACO V control system with a (then, only recently introduced) Allen-Bradley SLC500 PLC equipped with a half dozen 1747-NT4, 4 channel thermocouple input modules. Temperature measurements drifted by a degree or so, some more than others, and apparently at random, but closer investigation revealed it depended on how many zones were simultaneously calling for heat. Turned out the -NT4 module was more sensitive to common mode noise than the MACO V (several years later A-B introduced an -INT4 module with better isolation), and a quick fix was to swap out the original grounded thermocouple probes for ungrounded ones. Another option was to replace all the TC extension wire runs with shielded TC extension cable, ground the shield at the TC plug negative connection (in the process introducing an ongoing maintenance issue), and bond the negative TC extension wire at the module's analog common.

TC input wiring

It was a long walk to get here, but most other manufacturers also recommend bonding a grounded thermocouple's cable shield to the source (TC) rather than at the instrument side. The cable shield for an ungrounded TC is grounded at the input module.

Ungrounded thermocouples aren't a panacea, and don't necessarily help with noise. Many TC instrumentation amps connect the inverting input to ground through a high resistance. It they don't, any charge that builds up on the wires through op amp reverse bias leakage, static, etc. has nowhere to go, and affects measurements.

Figure 6 was borrowed from an excellent Analog Devices technical article, "Two Ways to Measure Temperature Using Thermocouples Feature Simplicity, Accuracy, and Flexibility" regarding their TC amplifiers.

In the OP's situation I'd bet there is a screw within reach of the TC ground wire with witness marks indicating prior connection.