Thank you @TheTias for the background info. This isn’t a user preference issue but an issue of poor design. The issue first arose because of lack of information coming from Blizzard. Separation of foum sub-sites does not mean separation of all sub-sites. In fact from limited testing I did it seems the acct. management site and shop are in the same sandbox.
Because the surface level look and feel of the new website implementation following the steps in the OP should have lead to specific outcome. It did not. From our discussion in the general forum, the new websites implementation uses 2 similar but different logins. Multi-login designs are flawed.
@jambrix:
The extra click of the login button was an unexpected and undocumented requirement of the new website implementation based on the info. coming from Thetias.
So, is the separated login/out capability of the forums and account mngt sites a design feature or an error in the implementation? If the answer is yes then it would be smart to inform the customer base of how the separation works between the shop, forums, and account management with respect to multiple logins. The old website implementation used a single sign-on such that logging out of 1 part of the site meant being logged out of the parts. Most sites and apps do not use multiple logins. Many sites/apps employ extra forced credentials checks when entering certain sensitive areas of the site/apps to protect from compromise.