Well, Instagram changed the response on their Oembed service sometime in December 2014.
And WordPress caches all oembed lookups the first time they’re displayed.
This means that all of your little blog posts which had an Instagram image or video (and more than 1 pageview) before December 2014 will look different than all your new posts. There’s no simple way of making all new embeds look old. But the other way around is easy.
1. Open your PhpMyAdmin or similar tool
2. Select the database WordPress is using
3. Click SQL and run this oneliner:
DELETE FROM `wp_postmeta` WHERE `meta_key` LIKE '%oembed_%'
Reload your WordPress site and voilá. That should do it.