Player Name on Network - UNet - Bug

I'm using this code to sync nickname of players across the network, using UNet, and it finally works (several searches and tries before getting something simple/working) \o/

However I noticed that if one client connects to the server directly after that it started (1~2 sec after), the first player copies the name of the second coming.

Is there someone that could help me to figure out why it behaves that way please?

 using UnityEngine;
 using UnityEngine.Networking;
 using System.Collections;
 
 [NetworkSettings (sendInterval=1)]
 public class PlayerNameSync : NetworkBehaviour {
 
     [SyncVar] // (hook = "WhenPlayerNameChange")]
     public string playerName = "";
     private Transform myTransform;
     public bool alreadyChanged = false;
 
     public override void OnStartLocalPlayer()
     {
         TransmitName();
         SetName();
     }
 
     void Awake()
     {
         myTransform = transform;
     }
 
     void FixedUpdate()
     {
         if (!alreadyChanged)
         {
             if (playerName != "")
             {
                 alreadyChanged = true;
                 myTransform.GetComponentInChildren<TextMesh>().text = playerName;
             }
         }
     }
 
     [Command]
     void CmdProvideNameToServer(string name)
     {
         playerName = name;
     }
 
     [Client]
     void TransmitName()
     {
             if (PlayerPrefs.HasKey("nickname"))
                 CmdProvideNameToServer(PlayerPrefs.GetString("nickname"));
             else
                 CmdProvideNameToServer("None");
     }
 
     void SetName()
     {
         if (!isLocalPlayer)
         {
             Debug.Log("SetName: " + playerName);
             myTransform.GetComponentInChildren<TextMesh>().text = playerName;
         }
         else
         {
             if (PlayerPrefs.HasKey("nickname"))
                 Debug.Log("nickname Registered: " + PlayerPrefs.GetString("nickname"));
             else
                 Debug.Log("nickname not Registered: " + playerName);
             myTransform.GetComponentInChildren<TextMesh>().text = PlayerPrefs.GetString("nickname");
         }
     }
 }