>>If you only have SSMS on that server, then it's OK. Just verify SQL Server version using
>>

>>select @@Version, SERVERPROPERTY('Edition') AS Edition,
>>	SERVERPROPERTY('ProductVersion') AS ProductVersion,
>>	SERVERPROPERTY('ProductLevel') AS ProductLevel

>
>On the server where the engine is installed, it gives this:
>
>Microsoft SQL Server Management Studio 10.50.1600.1
>Microsoft Analysis Services Client Tools 10.50.1600.1
>Microsoft Data Access Components (MDAC) 6.1.7601.17514
>Microsoft MSXML 3.0 6.0
>Microsoft Internet Explorer 8.0.7601.17514
>Microsoft .NET Framework 2.0.50727.5448
>Operating System 6.1.7601
>
>On the server where the engine is installed, your command gives this:
>
>Microsoft SQL Server 2008 R2 (RTM) - 10.50.1600.1 (X64) Apr 2 2010 15:48:46 Copyright (c) Microsoft Corporation Enterprise Edition (64-bit) on Windows NT 6.1 X64 (Build 7601: Service Pack 1) (Hypervisor)
>Enterprise Edition (64-bit)
>10.50.1600.1
>RTM
>
>On the server where the engine is installed, SP1 is then installed. However, what would then have to be done in order to assure that the SSMS tool on that server is up to date?

The first info is not from this command. However, the second output clearly shows you're using RTM version of SQL Server. So, I would install SQL Server 2008 R2 SP1 on that server.