In the US, Wechsler is a popular test, WAIS and Memory versions. MMPI-II, Minnesota Multiphasic Personality Inventory-II is also part of the battery. The Halstead Reitan brain injury battery is common. There are newer computerized test batteries, too.

btw, A battery is a set of various tests oriented toward a specific condition. There are more than 100 different tests to choose from to fit into a battery.

Here is a wiki about the Halstead-Reitan Battery
Halstead-Reitan Neuropsychological Battery - Wikipedia

If I remember, you are in Italy. There may be specific tests that are common to Italy.

Do you think that the Halstead-Retain Battery would be adequate to diagnosis my kind of cognitive issues?

It sure would be heading in the right direction. One of the newer computerized batteries with just a few directly administered tests could also help.

The point is that is does not sound like you had a Neuro Psychological Assessment of any value. Getting properly assessed with any appropriate battery will be worthwhile.