Eunice here from Shopify. Thanks for posting, and welcome to the Community!
I understand you’d like to transfer your original myshopify.com URL (store name) from your 1st store to your 2nd store. However, it’s not possible to do so at this time. Allmyshopify.com URLs stay with the store and cannot be transferred. Nor can they be permanently erased.
There is an active feature request to get this changed so that merchants can transfer their myshopify.com URLs between stores. I have gone ahead and added your feedback to this request.
It’s good to note that you can still purchase a custom domain, which is a domain that would likely end in .com or .co. For example, you could purchase admin2022.com if it is available. Learn more about custom domains here.
I think I’m a bit confused with how “store name” and custom domain relate?
For example, I opened a trial session for the store with a name “Glaxon Australia”.
I have purchased a domain for it even earlier (www.glaxon.com.au), but I realized the store had been set up under a wrong account before I connected the domain with the store.
Now I’m trying to open the store with same name under another account, and I am told that the store already exists at the beginning of store setup, even tho no domain is attached yet?
Also, are domain name and shopify “store name” both searchable in search engine?
You can change your store name by going to General > Settings from your Shopify admin. However, this isn’t to be mistaken with changing the myshopify.com URL; this is something that cannot be changed. It is something that can also not be erased, which is why you are seeing the ‘store already exists’ error message when you attempt to open up a new store using the same name (same myshopify.com URL).
A domain is the URL or web address where your customers go to find your store online. By default, your online store was given a myshopify.com URL when you signed up for Shopify. If you want to change the web address that’s displayed to customers while they browse your online store, then you need to use a custom domain.
With regards to what is searchable - this would be whatever your domain is. If you have purchased a domain then this would be searchable. If you haven’t purchased a domain then your default myshopify.com URL (store name) would be searchable.
Let me know if you need any further help with this. Domains can be tricky things!