So Micropub is the open standard that allows you to use a wealth of clients (https://indieweb.org/Micropub/Clients) to publish to your site. Twitter -> site Webmentions can be done with Bridgy. If you want to publish to Twitter from your site, you'll want something like https://remysharp.com/2019/06/18/send-outgoing-webmentions to automate sending the webmentions that will then trigger Bridgy to publish the post (https://martymcgui.re/2020/07/15/what-we-talk-about-when-were-talking-about-webmentions/ is also worth a read)