>CTRL+SHIFT+R will refresh the cache for intellisense, theOtherServer\theOtherInstance would be exactly what you have in your linked server, expand the "Linked Servers" item on object explorer and the name diplayed there is the one you need to type.
>
>[Update]I did not mentioned it, but if you are using "theOtherServer\theOtherInstance" (ie with the backslash) then you must enclose it in square brackets as I put it in the original query

I still do not understand the two word terminology here in the syntax. If you look at the attached file, you will see that I have defined a linked server. There is only one name in there. It is hidden. Lets called it MyOtherServer. So, in the syntax, I have something like this:

UPDATE Client
 SET Address=o.Address FROM Client m
 JOIN MyOtherServer.MyDatabase.Client o on o.Numero=m.Numero