I did some simple test and I got it!!!

Here is the code you pasted here (I added a grid to ensure that the connection is good and I changed the database).

Option Explicit

Private Logincn As ADODB.Connection
Private mrstData As ADODB.Recordset

Private Sub Command1_Click()
    Set Logincn = New ADODB.Connection
    With Logincn
    .Provider = "Microsoft.Jet.OLEDB.3.51"
    .ConnectionString = "D:\Projets VB\Nwind.Mdb"
    .Open
    End With
    
    Set mrstData = New ADODB.Recordset
    mrstData.Open "SELECT * FROM Employees", Logincn, adOpenStatic, adLockReadOnly
    Set DataGrid1.DataSource = mrstData
End Sub

Private Sub Command2_Click()
    Set Logincn = New ADODB.Connection
    With Logincn
    .Provider = "Microsoft.Jet.OLEDB.4.0"
    .ConnectionString = "D:\Projets VB\Nwind.Mdb"
    .CursorLocation = adUseClient
    .Open
    End With

    Set mrstData = New ADODB.Recordset
    mrstData.Open "SELECT * FROM Employees", Logincn, adOpenStatic, adLockReadOnly
    Set DataGrid1.DataSource = mrstData
End Sub

The only thing that was missing in your "New Code" was

.CursorLocation = adUseClient

I don't know why but his seems to have an importance with this version of the provider!