Sorry, but this does not address the issue outlined in my original post.
As I stated - I have attempted both the A record and CNAME record creation. Prior to posting here, I did a search on articles and help related to the creation of DNS records in Office365 as well as searched these forums. It seems that the subject line "web and DNS" is immediately lined to these articles which are really just a "how to create an A record" and "How to create a CNAME record". As I stated in my post - I already attempted this.
The concern is that I need to do web-forwarding to a URL. The URL is formatted as per my original post.
I can create the A record to an IP where my site is hosted - no problem.
I cannot seem to create a CNAME formatted http:\\www.mysight.com\~subdirectory
It gives me errors about needing to be greater than 3 characters (which it is).
Also, the redirect to the generic DNS configuration information is also incorrect and not working - there are several comments to that affect.