Date display differences in node templates
The proper thing to do here is to configure your field output to use the proper formatter.  On the Manage Display for your content type, eg admin/structure/types/manage/page/display, change the formatter for your field_event_start_date field to Custom, then click on the gear and change the format to 'g:ia'.  You should then be able to use {{ content.field_event_start_date }} in your template.

The standard Twig date() filter doesn't use the date.formatter service, so timezones and localization settings will not be honored.

TwigExtension defines a custom filter for Drupal that will call out to DateFormatter::format() called format_date().  This should work

{{ node.field_event_start_date.value|format_date("custom", "g:ia") }}


This apparently only works for Timestamp fields and not datetime or daterange fields.