Minecraft – testfor lapis lazuli anywhere in inventory

When testing for pre-existing data, you must declare the data as-is. The item ID is saved with a namespace, defaulting to minecraft, but your input via /testfor (or /scoreboard) will not be automatically corrected. As such, you must input the namespace that the item is saved with, becoming minecraft:potion.

The Damage tag is saved with a tag-type of Short. In order to declare a Short, you must append the numerical value with an 's'.

Your command will become:

/testfor @e[type=ThrownPotion] {Potion:{id:"minecraft:potion",Damage:16388s}}

When accessing pre-existing NBT data, you have to specify it exactly as it's stored. This includes the name, value, and the datatype. The Slot tag is stored as a Byte, while you've declared it as an Integer. You can append the numerical value with a "b" to declare it as a Byte:

/testfor @a {Inventory:[{Slot:0b,id:"minecraft:wooden_button"}]}

To mention, while the command will output whether or any there is any player at all with a wooden button in that slot, it does not allow you to target that player afterwards.

You can instead assign a label based on the player's NBT data and then target them based on that label:

1.8:

Objective required.

/scoreboard objectives add HasItem dummy

Detection commands.

/scoreboard players set @a HasItem 0
/scoreboard players set @a HasItem 1 {Inventory:[{Slot:0b,id:"minecraft:wooden_button"}]}
/say @a[score_HasItem_min=1]

1.9:

No objective needed, only detection commands.

/scoreboard players tag @a[tag=HasItem] remove HasItem
/scoreboard players tag @a[tag=!HasItem] add HasItem {Inventory:[{Slot:0b,id:"minecraft:wooden_button"}]}
/say @a[tag=HasItem]