A thermocouple uses two different types of metal to form a junction that creates a voltage potential that has a known relationship to temperature.
The K-Type thermocouple uses alloys called Chromel & Alumel (Nickel Chromium & Nickel Aluminium).
To avoid errors, you want to use the same type of alloys in your wiring all the way to your measuring device (DMM, etc). The connectors are also made with these alloys.
On the probe you show, I imagine the two centre screws are for mechanical assembly. Two outer screws connect to the wires from inside the tube. The other two outer screws are for your extension cable. Be careful with disassembly and reassembly as you cannot easily tell the type of alloy of the metal contacts, so you don't want to get the mixed up. I think I can see plus and minus symbols on those connection points.
You can buy the connectors and extension cable from usual suppliers. I bought some from element14 under the brand name Lab Facility.
Also note that there are different standards for colour coding of both connectors and cable insulation - IEC, ANSI, DIN, and JIS. Your yellow connectors for K-Type are actually the ANSI (US) standard, and you might want the green connectors and cables per IEC/DIN standards.
Here are some links to PDFs with further info
https://assets.omega.com/landing-pages/colorcodes/tc_colorcodes.pdfhttps://www.labfacility.com/media/productattach/t/h/thermocouple_product_guide_labfacility.pdfI've also one attached below
J-Type uses different alloys to K-Type, so also make sure you know what you have and don't mix them up.
Annoyingly the different standards do re-use some of the same colours for different thermocouple types.

You want to minimise the use of copper connections. If curious, you can learn more by reading up on isothermal blocks and cold junction compensation.