squarespace "built in" name shows up in Google search, not my custom name, when key words /tags are searched

Why does my "built in" name or assigned name (before I went live with my custom name) show up in Google when I search specific key words? (tags used in the blog portion of my site.) So the user who searches those terms will get the long two-word.squarespace name, not my 5 letter custom name, in use for decades.

TL DR: I want pages with specific key words / tags  to show up under my general website name

If your built-in domain is showing in search results, this means Google indexed that URL at some point. This can happen if the site is published before the custom domain is connected, but it sounds like you didn't publish until after that.

Now that your custom domain is connected, search results should update to reflect the correct URL over time.

To help speed things up, first make sure that you've added the keywords and tags you'd like associated with your site throughout your site's content. You'll need to do this mindfully though, so that you're writing for people and not bots. Squarespace has some good tips on this: Keyword best practices

When you're satisfied, submit a sitemap in Google Search Console to request a reindex: Verifying your site with Google Search Console - Index your site