Windows 7 cannot ping a name, only an IP address

Sorry about this, but all the similar stuff I have read does not solve my problem.

I have Windows 7 Professional on these two machines, and they are networked as public. The sharing is set to file sharing, discovery, etc. - all opened up. Windows Explorer is fine, and I can see the network great and the shared folders. I can go to a command prompt (as Administrator) and ping the other machine using the IP address.

I CANNOT ping it using its name. I want to do that to use Remote Desktop Connection. The workgroup names are the same, and I am a user and administrator on both machines with the same password.

I have tried with the Windows firewall down on both machines and the antivirus software down on both. I am using a Belkin router with WAN ping blocking but no LAN options. Both machines show just fine on the router.

Three days have passed, and I cannot crack this problem. Last year I used Windows XP, and it navigated by name perfectly. Is it Windows 7 Professional doing something differently?

Diagnostics results

Microsoft Windows [Version 6.1.7601]
Copyright (c) 2009 Microsoft Corporation.  All rights reserved.

C:\Windows\system32>ping phoenix

Pinging phoenix.Belkin [198.105.251.24] with 32 bytes of data:
Request timed out.
Request timed out.
Request timed out.
Request timed out.

Ping statistics for 198.105.251.24:
    Packets: Sent = 4, Received = 0, Lost = 4 (100% loss),

C:\Windows\system32>ipconfig /all

    Windows IP Configuration

       Host Name . . . . . . . . . . . . : october7
       Primary Dns Suffix  . . . . . . . :
       Node Type . . . . . . . . . . . . : Hybrid
       IP Routing Enabled. . . . . . . . : No
       WINS Proxy Enabled. . . . . . . . : No
       DNS Suffix Search List. . . . . . : Belkin

    Ethernet adapter Local Area Connection 2:

       Connection-specific DNS Suffix  . : Belkin
       Description . . . . . . . . . . . : Realtek PCIe GBE Family Controller
       Physical Address. . . . . . . . . : 6C-62-6D-69-B4-6B
       DHCP Enabled. . . . . . . . . . . : Yes
       Autoconfiguration Enabled . . . . : Yes
       Link-local IPv6 Address . . . . . : fe80::a815:ffa0:1230:86fc%13(Preferred)
       IPv4 Address. . . . . . . . . . . : 192.168.2.10(Preferred)
       Subnet Mask . . . . . . . . . . . : 255.255.255.0
       Lease Obtained. . . . . . . . . . : Tuesday, July 23, 2013 5:08:44 PM
       Lease Expires . . . . . . . . . . : Friday, August 29, 2149 11:47:41 PM
       Default Gateway . . . . . . . . . : 192.168.2.1
       DHCP Server . . . . . . . . . . . : 192.168.2.1
       DHCPv6 IAID . . . . . . . . . . . : 292315757
       DHCPv6 Client DUID. . . . . . . . : 00-01-00-01-17-48-BC-21-6C-62-6D-69-B4-6B

       DNS Servers . . . . . . . . . . . : 192.168.2.1
       NetBIOS over Tcpip. . . . . . . . : Enabled

    Tunnel adapter isatap.Belkin:

       Media State . . . . . . . . . . . : Media disconnected
       Connection-specific DNS Suffix  . : Belkin
       Description . . . . . . . . . . . : Microsoft ISATAP Adapter
       Physical Address. . . . . . . . . : 00-00-00-00-00-00-00-E0
       DHCP Enabled. . . . . . . . . . . : No
       Autoconfiguration Enabled . . . . : Yes

    Tunnel adapter Local Area Connection* 11:

       Connection-specific DNS Suffix  . :
       Description . . . . . . . . . . . : Teredo Tunneling Pseudo-Interface
       Physical Address. . . . . . . . . : 00-00-00-00-00-00-00-E0
       DHCP Enabled. . . . . . . . . . . : No
       Autoconfiguration Enabled . . . . : Yes
       IPv6 Address. . . . . . . . . . . : 2001:0:9d38:6ab8:34e2:129c:b8ad:7db(Preferred)
       Link-local IPv6 Address . . . . . : fe80::34e2:129c:b8ad:7db%11(Preferred)
       Default Gateway . . . . . . . . . : ::
       NetBIOS over Tcpip. . . . . . . . : Disabled

I've noticed name resolution is a hit or miss function on a peer-to-peer network with no central DNS server (on any version of Windows).

You may check to see if the zeroconf networking service is running which is supposed to resolve names and a host of other things.

I usually install the Bonjour service from Apple which works flawlessly for local name resolution on small networks.