I'm not quite sure what the pictures are about, but PEX tubing is normally used for water supply line. In most cases, when it freezes, the line itself is okay, but the fittings split. Once you have favorable weather conditions, you can (carefully) pressure test it with air. For repair, you will need PEX fittings, and there is a special crimp tool for the crimp bands.
PEX is available at your local home improvement store, as well as the tool.
Avoid 'quick' type fittings, like Sharkbite, as they tend to get loose with movement.
Edit: If it is only the supply line coming from a quick connect compression type fitting to the faucet, then yes, common flex supply line is what you want. Inspect the outdoor faucet for leaks, that is subject to cracking.
Doug and Patience and Shelby the mutt
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