I am in the middle of updating to https. I changed our DNS A record to point to the new ip that was provided by the host on Oct 31st. Now when I google our company name, the first entry is our website at www.rainstor.com. They have the same ip address according to Whois. My web hosting company says it is not their problem that the other company is pointing to our website! Can anyone help? Has anyone had this problem before? If so, who is responsible, hosting or rainstor?
To answer the question, "who is responsible", the only people who can change the DNS record for rainstor would be whoever has control of their DNS records. It might be rainstor directly or it might be a subcontractor or service provider of theirs.
I don't really see it as a problem that they haven't updated their DNS records. It really is their problem and not yours. I can see it's a bit of a nuisance for you, but the important thing is you have your A record setup correctly and it points to the server that you control.
There are measures you can take to prevent your content from appearing on their domain:
configure your server to send back an error for any address that matches a URL that doesn't contain your correct domain. The actual message could be a '505 server error' or '404 not found' or '401 not authorized' or '301 moved' (which will redirect to a different URL of your choosing). Basically, anything you want.
use a Rewrite Rule, or
use a Virtual Host configuration, or
write code into the sites PHP (if that's what you're using) to deliver appropriate headers
The actual solution will depend on what server software you are using.
The DNS A record governs where all requests go, everything else is irrelevant. After a client requests your page, that request is handled by the destination server at the IP address you set the A record to. That destination server configuration determines what type of connection(s) you support - HTTP, HTTPS, and optionally a redirect to one or the other type.
What may be confusing things is DNS propagation time. You can test for yourself at a site like whatsmydns.net to see which A records are being served (you very well may find the old IP address is still being returned for a day or two).
You can safely ignore anything to do with the old hosting company now that you have switched to HTTPS and a new host. Very soon all requests for your site will be directed to the new IP address you set with the new DNS A record.
One tiny thing to double-check, make certain you only have one DNS A record. It is technically possible to use multiple but this is probably not what you want, especially if the second record is still pointing at the old host's IP address.