There are a number of issues.
You are visiting store.mydomain.com in your browser, but the Linode DNS manager is configured for store.myurl.com.
The nameservers for both mydomain.com and myurl.com are set to nameservers other than the ones operated by Linode, so the records you specify in the Linode DNS manager have no effect.
You have created a domain zone for the subdomain store.myurl.com, which is probably not necessary, especially if you're also using Linode's DNS manager to host the zone for myurl.com. If you wish to create a CNAME record for store.myurl.com, you can not add it as a separate domain zone in the manager since you are then also required to have NS records for store.myurl.com, and you can not simultaneously have a CNAME record and NS records (or any other records) for a particular hostname.
You have added A records for store.myurl.com and shop.store.myurl.com that point to an IP address that is reserved for local identification per RFC 5735.
You have added a CNAME record for
www.store.myurl.com that points to store.theirurl.com, but a query for store.theirurl.com returns NXDOMAIN. In theory, if store.theirurl.com existed and was running a web server, visiting
www.store.myurl.com in your browser would result in the request being served by store.theirurl.com, but you want to achieve this effect for store.mydomain.com, not
www.store.myurl.com.
If you are using the Linode DNS manager to host the zones for both myurl.com and store.myurl.com, you should most likely delete both of those and add a single domain zone for only mydomain.com. The add a CNAME record with a hostname of "store" pointing to mydomainname.shopify.com (or was it store.theirurl.com?).
Good luck!