Difference between Make a claim for something and Make a claim to something

make a claim for something—demand compensation payment (under an insurance contract):

• If you send a parcel that becomes lost or damaged … you may wish to make a claim for compensation.

make a claim to something—(also: make a claim on something)

1. assert one’s right to smth.:

• Japan made a claim to this territory and argued that it was illegally occupied.

2. allege or aver smth.:

• As I am not an anthropologist, sociologist or historian, I make no claim to understanding culture from any of those perspectives.