The Dwarf Subrace, House Kundarak has a an ability called Warder's Intuition. It goes as follows:

Warder’s Intuition

When you make an Intelligence (Investigation) check or an ability check using thieves' tools, you can roll a d4 and add the number rolled to the ability check.

I read this two different ways:

You get to add a d4 roll to every Investigation check. You also get to add a d4 roll to every ability check as long as you use thieve's tools.

You get to add a d4 roll to every Investigation or ability check as long as you use thieve's tools.

Its probably RAI the second one, but would the first one work as well?

It's the first one because of where the "or" is. It's written [Investigation check] or [Thieves' Tools ability check].

If it were the second it would just say "any ability check made using thieves' tools" because Investigation would be implicitly included.

That's what I was hoping. Thank you.

An “Investigation check” is an ability check. It’s an “Intelligence (Investigation) Check.” There are no “skill checks” in D&D, there are only Ability checks. Ones that specify a (Skill) or tool are just Ability Checks that you can add your proficiency bonus to if you are proficient with the skill or tool referenced. People colloquially call them “skill checks” but that is in fact a misnomer.

How do you configure a tool check with Thieves Tools to reflect this extra d4?

Quote from JonJones >>

How do you configure a tool check with Thieves Tools to reflect this extra d4?

Here are some relevant rules:

Proficiency with a tool allows you to add your proficiency bonus to any ability check you make using that tool. Tool use is not tied to a single ability, since proficiency with a tool represents broader knowledge of its use. For example, the DM might ask you to make a Dexterity check to carve a fine detail with your woodcarver's tools, or a Strength check to make something out of particularly hard wood.

and

Proficiency with these [Thieves'] tools lets you add your proficiency bonus to any ability checks you make to disarm traps or open locks.

and

Other Dexterity Checks

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

• Pick a lock
• Disable a trap

So, in this case the way to do it is to ask for a straight Dexterity Check (not associated with a specific skill) which factors in your Dexterity modifier and then if the player is proficient with the Thieves' Tools they add their proficiency bonus.  Then, The Warder's Intuition gives the player the option to add a d4 to that result.

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Now, for a more complex situation that may involve proficiency in more than one skill at once, or proficiency in a skill and proficiency in a tool at the same time, the standard rules unfortunately have this to say about that:

Your proficiency bonus can’t be added to a single die roll or other number more than once.

However, the alternate rules provided in Xanathar's Guide To Everything provide some rules that are more favorable:

To make tool proficiencies more attractive choices for the characters, you can use the methods outlined below.

 Advantage. If the use of a tool and the use of a skill both apply to a check, and a character is proficient with the tool and the skill, consider allowing the character to make the check with advantage.

Added Benefit. In addition, consider giving characters who have both a relevant skill and a relevant tool proficiency an added benefit on a successful check. This benefit might be in the form of more detailed information or could simulate the effect of a different sort of successful check. For example, a character proficient with mason’s tools makes a successful Wisdom (Perception) check to find a secret door in a stone wall. Not only does the character notice the door’s presence, but you decide that the tool proficiency entitles the character to an automatic success on an Intelligence (Investigation) check to determine how to open the door.

For the Thieves' Tools in particular, some suggested skill / tool combos include:

History / Thieves' Tools:   "Your knowledge of traps grants you insight when answering questions about locations that are renowned for their traps."

Investigation / Thieves' Tools and Perception / Thieves' Tools:   "You gain additional insight when looking for traps, because you have learned a variety of common signs that betray their presence."

Hope that helps!