$(…).wpcf7InitForm() is not a function

so… one solution could be to wrap the whole first self-invoked function into a wpcf7.init() function:

wpcf7.init = function() {

	$( function() {
		wpcf7.supportHtml5 = ( function() {
			var features = {};
			var input = document.createElement( 'input' );

			features.placeholder = 'placeholder' in input;

			var inputTypes = [ 'email', 'url', 'tel', 'number', 'range', 'date' ];

			$.each( inputTypes, function( index, value ) {
				input.setAttribute( 'type', value );
				features[ value ] = input.type !== 'text';
			} );

			return features;
		} )();

		$( 'div.wpcf7 > form' ).each( function() {
			var $form = $( this );

			$form.submit( function( event ) {
				if ( typeof window.FormData !== 'function' ) {
					return;
				}

				wpcf7.submit( $form );
				event.preventDefault();
			} );

			$( '.wpcf7-submit', $form ).after( '<span class="ajax-loader"></span>' );

			wpcf7.toggleSubmit( $form );

			$form.on( 'click', '.wpcf7-acceptance', function() {
				wpcf7.toggleSubmit( $form );
			} );

			// Exclusive Checkbox
			$( '.wpcf7-exclusive-checkbox', $form ).on( 'click', 'input:checkbox', function() {
				var name = $( this ).attr( 'name' );
				$form.find( 'input:checkbox[name="' + name + '"]' ).not( this ).prop( 'checked', false );
			} );

			// Free Text Option for Checkboxes and Radio Buttons
			$( '.wpcf7-list-item.has-free-text', $form ).each( function() {
				var $freetext = $( ':input.wpcf7-free-text', this );
				var $wrap = $( this ).closest( '.wpcf7-form-control' );

				if ( $( ':checkbox, :radio', this ).is( ':checked' ) ) {
					$freetext.prop( 'disabled', false );
				} else {
					$freetext.prop( 'disabled', true );
				}

				$wrap.on( 'change', ':checkbox, :radio', function() {
					var $cb = $( '.has-free-text', $wrap ).find( ':checkbox, :radio' );

					if ( $cb.is( ':checked' ) ) {
						$freetext.prop( 'disabled', false ).focus();
					} else {
						$freetext.prop( 'disabled', true );
					}
				} );
			} );

			// Placeholder Fallback
			if ( ! wpcf7.supportHtml5.placeholder ) {
				$( '[placeholder]', $form ).each( function() {
					$( this ).val( $( this ).attr( 'placeholder' ) );
					$( this ).addClass( 'placeheld' );

					$( this ).focus( function() {
						if ( $( this ).hasClass( 'placeheld' ) ) {
							$( this ).val( '' ).removeClass( 'placeheld' );
						}
					} );

					$( this ).blur( function() {
						if ( '' === $( this ).val() ) {
							$( this ).val( $( this ).attr( 'placeholder' ) );
							$( this ).addClass( 'placeheld' );
						}
					} );
				} );
			}

			if ( wpcf7.jqueryUi && ! wpcf7.supportHtml5.date ) {
				$form.find( 'input.wpcf7-date[type="date"]' ).each( function() {
					$( this ).datepicker( {
						dateFormat: 'yy-mm-dd',
						minDate: new Date( $( this ).attr( 'min' ) ),
						maxDate: new Date( $( this ).attr( 'max' ) )
					} );
				} );
			}

			if ( wpcf7.jqueryUi && ! wpcf7.supportHtml5.number ) {
				$form.find( 'input.wpcf7-number[type="number"]' ).each( function() {
					$( this ).spinner( {
						min: $( this ).attr( 'min' ),
						max: $( this ).attr( 'max' ),
						step: $( this ).attr( 'step' )
					} );
				} );
			}

			// Character Count
			$( '.wpcf7-character-count', $form ).each( function() {
				var $count = $( this );
				var name = $count.attr( 'data-target-name' );
				var down = $count.hasClass( 'down' );
				var starting = parseInt( $count.attr( 'data-starting-value' ), 10 );
				var maximum = parseInt( $count.attr( 'data-maximum-value' ), 10 );
				var minimum = parseInt( $count.attr( 'data-minimum-value' ), 10 );

				var updateCount = function( target ) {
					var $target = $( target );
					var length = $target.val().length;
					var count = down ? starting - length : length;
					$count.attr( 'data-current-value', count );
					$count.text( count );

					if ( maximum && maximum < length ) {
						$count.addClass( 'too-long' );
					} else {
						$count.removeClass( 'too-long' );
					}

					if ( minimum && length < minimum ) {
						$count.addClass( 'too-short' );
					} else {
						$count.removeClass( 'too-short' );
					}
				};

				$( ':input[name="' + name + '"]', $form ).each( function() {
					updateCount( this );

					$( this ).keyup( function() {
						updateCount( this );
					} );
				} );
			} );

			$form.on( 'change', '.wpcf7-validates-as-url', function() {
				var val = $.trim( $( this ).val() );

				// check the scheme part
				if ( val && ! val.match( /^[a-z][a-z0-9.+-]*:/i ) ) {
					val = val.replace( /^\/+/, '' );
					val = 'http://' + val;
				}

				$( this ).val( val );
			} );

			if ( wpcf7.cached ) {
				wpcf7.refill( $form );
			}
		} );
	} );

}

wpcf7.init();

Then later, I can call this from my own code:

if( typeof wpcf7 === 'object' && typeof wpcf7.init === 'function' ) {
  wpcf7.init();
}

@konsument, I uploaded the full rewritten scripts.js file to my server, you can download it from there and replace the file /plugins/contact-form-7/includes/js/scripts.js with my version:

http://test.rassohilber.com/wpcf7/scripts.js