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]
?>

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".

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 $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.