I'm confused with the definition of things.
In my view High Availability is the property of a system that aims to ensure an agreed level of operational performance and usually is expressed as a percentage of uptime in a given year.
On another side Availability in CAP theorem state that in a distributed system, reads and writes would always succeed even when network connectivity is unavailable between nodes.
Therefore I assumed that these two things are absolutely different.
If this true, could we say that the distributed system may be High Available and don't have a property of Availability simultaneously in terms of CAP and vice versa?
You are very much correct in your understanding that the terms High Availability (which is used in the context of Cluster Solutions) and Availability (which is used in the context of CAP theorem to describe Accessibility of the data stored in the distributed system) are completely different.
In the CAP theorem, the term Partitioning (P of CAP) refers to High Availability.
Yes, a distributed system can possess High Availability (i.e. Partitioning) and not have Availability (i.e. Accessibility) or vice versa. In this scenario the system can achieve Consistency (C of CAP) in both the cases.