Hi Guys,
timezone-identifiers available from @lat instructions are listed at the end. The important for my case is Asia/Nicosia, Europe/Athens, which are available.( Running Joomla 4 system report, on same PHP, it shows also Europe/Nicosia and UTC as default timezone. Joomla has no problem with the PHP 8.1.1).

The actual timezone declared in php.ini is date.timezone = "Europe/Athens"

The Intl extension is also available.
I entered also in php.ini
[intl]
intl.default_locale = el_GR.UTF-8 which it had no effect on the issue.

Attached also screenshot From info.php
Click image for larger version.  Name: Screenshot 2024-05-13 at 6.25.13 PM.jpg  Views: 78  Size: 51.4 KB  ID: 20608

Asia/Aden, Asia/Almaty, Asia/Amman, Asia/Anadyr, Asia/Aqtau, Asia/Aqtobe, Asia/Ashgabat, Asia/Atyrau, Asia/Baghdad, Asia/Bahrain, Asia/Baku, Asia/Bangkok, Asia/Barnaul, Asia/Beirut, Asia/Bishkek, Asia/Brunei, Asia/Chita, Asia/Choibalsan, Asia/Colombo, Asia/Damascus, Asia/Dhaka, Asia/Dili, Asia/Dushanbe, Asia/Famagusta, Asia/Gaza, Asia/Hebron, Asia/Ho_Chi_Minh, Asia/Hong_Kong, Asia/Hovd, Asia/Irkutsk, Asia/Jakarta, Asia/Jayapura, Asia/Jerusalem, Asia/Kabul, Asia/Kamchatka, Asia/Karachi, Asia/Kathmandu, Asia/Khandyga, Asia/Kolkata, Asia/Krasnoyarsk, Asia/Kuala_Lumpur, Asia/Kuching, Asia/Kuwait, Asia/Macau, Asia/Magadan, Asia/Makassar, Asia/Manila, Asia/Muscat, Asia/Nicosia, Asia/Novokuznetsk, etc...., Europe/Amsterdam, Europe/Andorra, Europe/Astrakhan, Europe/Athens, Europe/Belgrade, etc....