[RPG] What are the special pitons which come in a climber’s kit

It's not explicitly specified which ability you would use to employ these tools. Ultimately, it's up to your DM which ability is used for any ability check, and the default matching of skills to abilities is only a default. Both the Player's Basic Rules and the PHB include this section in Chapter 7:

Variant: Skills with Different Abilities

So it is up to your DM to decide which ability or skill applies to any ability check you make. With tool-based ability checks, this is even more the case, since they don't have the defaults that skills come with.

However, the 2 tool proficiencies in the question are both included in the list of examples of potential Intelligence checks.

Other Intelligence Checks.

The DM might call for an Intelligence check when you try to accomplish tasks like the following:

• Communicate with a creature without using words
• Estimate the value of a precious item
• Pull together a disguise to pass as a city guard
• Forge a document
• Recall lore about a craft or trade
• Win a game of skill

So the developers must have thought using these tools would generally be Intelligence checks.

The rules for the climbing kit already describe how to use the gear:

You can use the climber's kit as an action to anchor yourself; when you do, you can't fall more than 25 feet from the point where you anchored yourself, and you can't climb more than 25 feet away from that point without undoing the anchor.

The pitons and harness are already encompassed in those rules. If you want to create other rules for them then that's up to the DM.