[dba-SQLServer] Registering SQL Servers

I have 3 computers with SQL Server installed, full install.  I want to
register the databases on each machine on each other machine so that I can
see the three databases from EM regardless of which machine I am on.

I will admit up front that I did the install under different user names,
i.e. under the default Administrator on one and under jcolby on the other
two (I think).  

I am using windows authentication.  I have a "reasonably strong" 10
character pneumonic password (first character of 10 words, but no numbers or
special characters).  I have created identical Administrator / password
accounts on all three machines.  All three machines can see shares from each
other without entering any passwords / user names (as long as I'm logged in
as administrator on that machine).

I have logged off and back on as the administrator using the identical
password on each machine.

Neo1 can see Local and can see (and register) Soltek, but if I try and
register Neo2 I can see the server but get ""Neo2 - Login failed for
'Neo2\Guest'.
Neo2 can see Local and can see (and register) Soltek, but if I try and
register Neo1 I can see the server but get the same "login failed for
Neo1\Guest" error.
Soltek can see Local (although it is CALLED SOLTEK!!!  It can see Neo1 and
Neo2 but cannot register them, getting the same "login failed" message.

What is going on here.  How do I sync them up so that all three can see and
register the other two.

Also what is the difference between the solid green circle with a white
arrow (in the server group tree) and the white circle with the green arrow?
Neo1 shows green with white arrows, Neo2 shows white with green arrows, and
Soltek shows its own name instead of Local and a solid green with white
arrow.

I believe Soltek was showing a Local but I couldn't connect to it, so I
deleted it and re-registered it to itself which is why I am not seeing
Local.

Can anyone briefly and succinctly explain what is happening during this
registration process, and how to get where AI want to go?

TIA,

John W. Colby
www.ColbyConsulting.com