you are authorized by someone to make payments

This could define who can make the decision whether or not to pay. Having authority does not necessarily require the one who holds it to exercise it in favor of a third party, although sometimes that is the case. The one with authority may, however, also have discretion, in which case the authority to determine the outcome also represents power.

Es geht m. E. ja nicht nur um die "Genehmigung", sondern mehr nohc um die Entscheidung ob/ob nicht bezahlt wird

in the sense of its compulsory character? (Hypothesis).

Of the long list of possibilities I'd imagine that:

Befugnis
Vollmacht
Ermächtigung

are the top 3 most likely candidates to have been (mis-) translated here, so feeding them into the same system DE-EN might help. Good luck with the detective work - sounds like quite a job to have landed...

It sounds like a mistranslation of the term "authority to pay" or A/P, meaning authorization (or discretion) to make payment.
In your case, it would mean that the parties with authority to make payment are the insurance companies and welfare departments. In other words, no other office is authorized to make payment.
You'll find plenty of examples of this on the Internet:

AUTHORITY TO PAY STATE AND LOCAL TAXES ON PROPERTY
AFTER ENTRY OF AN ORDER OF FORFEITURE

The Attorney General has discretionary authority to make payments of state and local tax claims against civilly forfeited property after a forfeiture order has issued, based on her equitable discretion to administer civilly forfeited property, under 21 U.S.C. § 881(b)-(e) and 28 U.S.C. § 524(c)(1).

OR (and I apologize about the depressing content):
"The Fiscal Year 1995 Appropriation Act for Interior and Related Agencies includes authority to pay funeral expenses for employees killed in the line of duty"

In English legalesque, eliminate the article ("the)