It should be noted that the locale string passed into IntlDateFormatter's constructor supports UCA keywords. So you can, for example, do things like this to output the year as a Japanese era year:

<?php
$now = new DateTime();    //DateTime is a core PHP class as of version 5.2.0

$formatter = new IntlDateFormatter('ja_JP', IntlDateFormatter::FULL,
        IntlDateFormatter::FULL, 'Asia/Tokyo', IntlDateFormatter::GREGORIAN);

echo 'It is now: "' . $formatter->format($now) . '" in Tokyo' . "\n";
//above gives [It is now: "2011年8月19日金曜日 23時32分27秒JST" in Tokyo]

$formatter = new IntlDateFormatter('ja_JP@calendar=japanese', IntlDateFormatter::FULL,
        IntlDateFormatter::FULL, 'Asia/Tokyo', IntlDateFormatter::TRADITIONAL);

echo 'It is now: "' . $formatter->format($now) . '" in Tokyo' . "\n";
//above gives [It is now: "平成23年8月19日金曜日 23時32分27秒JST" in Tokyo]
?>

Documentation says "timezone: Time zone ID, default is system default."

The "system default" really means only the "TZ" environment variable on Unix/Linux systems. It does not mean PHP ini setting or value set via date_default_timezone_set() or the OS default time zone in file "/etc/timezone".

The documentation says that $datetype and $timetype can also be NULL,  in which case ICUʼs default date type or time type will be used.

But when declare (strict_types=1); is also set, Intl fails to create the IntlDateFormatter class, and it returns an error "datefmt_create: unable to parse input parameters".

The $locale can although contain the information about the calendar

<?php
//...
        $traditionalFormatter  =  new IntlDateFormatter(
            $locale.'@calendar='.$calendar,
            IntlDateFormatter::SHORT,
            IntlDateFormatter::SHORT,
            'Europe/Berlin',
            IntlDateFormatter::TRADITIONAL,
            'yyyy/MM/dd HH:mm:ss' // ICU-datetime-format
        );
//..
?>

To get the allowed values for $calendar, use the following code:

<?php
$bundle=new ResourceBundle('','ICUDATA');
$cnames=[];
$calendars=$bundle->get('calendar');
foreach($calendars as $n=>$v){
    $cnames[]=$n;
}
echo (print_r($cnames,true));

?>

The result is:
Array
(
    [0] => buddhist
    [1] => chinese
    [2] => coptic
    [3] => dangi
    [4] => default
    [5] => ethiopic
    [6] => ethiopic-amete-alem
    [7] => gregorian
    [8] => hebrew
    [9] => indian
    [10] => islamic
    [11] => islamic-civil
    [12] => japanese
    [13] => persian
    [14] => roc
)

Just to be clear, to use any non-gregorian calendar:

a. The locale must be in the form of locale@calendar=calendar-name

b. The calendar must use the intlDateFormatter::TRADITIONAL constant, or an intlCalendar object created with a locale as specified in a.
 
The calendar name can be 'gregorian', even if the intlDateFormatter::TRADITIONAL constant is used, which makes for a more generic code approach.