You have a couple of conflicting arguments here, but what you want to do is completely possible.
To keep something out of the RSS feeds, and considering it really is a different type of content from your blog posts, I’d recommend a custom post type called “Links” or something.
The custom post type will have its own archive and RSS feed, separate from your posts.
To get it included in your home page feed, it depends on how your home page is configured, but if it’s just the “blog” archive, then it’s a pretty simple filter using pre_get_posts to include your new CPT into that archive.
For distribution, you could use Jetpack or other plugins to then auto-tweet new links of your CPT. That would then handle distribution, keep the link on your site, and capable of being perused or searched later.
Last thing, based on your note about tags, would be to enable support for tags and categories on your new custom post type. That would then include both posts and links (your CPT) when someone clicks the “Instagram” tag archive, or whatever.
This does include the code, but as you said you don’t need that, I think this answer should give you the roadmap you’re looking for; but to implement it properly it’s definitely going to take a little bit of intermediate code.