In the meantime, I was able to solve the problem by myself with the help of two links from the Internet (one of them from Stackoverflow).

The problem was due to the fact that at the time I updated the database, a commit flag was apparently not set internally.

When debugging, I could see that the DataSet had been updated correctly, but a flag was still missing, which meant that the changes were not written and no error message was displayed. This situation was very confusing.

Link 1: http://www.codingeverything.com/2013/01/firing-datagridview-cellvaluechanged.html

Link 2: DataGridView row is still dirty after committing changes

From link 1 I first took the use of the events, while link 2 provided a valuable tip regarding the use of a BindingSource and a Handler.

Since my problem already exists in an old message on Stackoverflow, which has not received a single answer, I would like to publish my solution to the problem here.

First the function dgvConfig which now uses a global BindingSource.

Private Sub dgvConfig(ByRef dgv As DataGridView)
    Try
        Dim col1 As New DataGridViewTextBoxColumn
        col1.Name = "col1"
        col1.HeaderText = "Host"
        col1.DataPropertyName = "myKey"
        Me.dgvRmServers.Columns.Add(col1)

        Dim col2 As New DataGridViewCheckBoxColumn
        col2.Name = "col2"
        col2.HeaderText = "Active"
        col2.DataPropertyName = "myValue"
        col2.TrueValue = "true"
        col2.FalseValue = "false"

        bs.DataSource = _dsxConfigDb.DataSet.Tables("hosts")

        Me.dgvRmServers.Columns.Add(col2)
        Me.dgvRmServers.AutoGenerateColumns = False
        Me.dgvRmServers.SelectionMode = DataGridViewSelectionMode.FullRowSelect
        Me.dgvRmServers.AllowUserToAddRows = False
        Me.dgvRmServers.AllowUserToDeleteRows = False             
        Me.dgvRmServers.DataSource = _bs

        AddHandler _bs.ListChanged, AddressOf CustomersBindingSource_ListChanged

    Catch ex As Exception

    End Try
End Sub

Next are the EventHandlers I used.

CustomersBindingSource_ListChanged:

Public Sub CustomersBindingSource_ListChanged(sender As Object, e As ListChangedEventArgs)
    Try
        _dsxConfigDb.SqliteUpdateKvp("hosts", _dsxConfigDb.DataSet.Tables("hosts").Rows(dgvRmServers.CurrentCell.RowIndex)("myKey"), dgvRmServers.Rows(dgvRmServers.CurrentCell.RowIndex).Cells(1).Value)
    Catch ex As Exception

    End Try
End Sub

DataGridView_CurrentCellDirtyStateChanged:

Private Sub DataGridView_CurrentCellDirtyStateChanged(sender As System.Object, e As EventArgs) Handles dgvRmServers.CurrentCellDirtyStateChanged
    If dgvRmServers.IsCurrentCellDirty Then
        dgvRmServers.EndEdit(DataGridViewDataErrorContexts.Commit)
    End If
End Sub

DataGridView_CellValueChanged:

Private Sub DataGridView_CellValueChanged(sender As DataGridView, e As DataGridViewCellEventArgs) Handles dgvRmServers.CellValueChanged
    If e.RowIndex = -1 Then
        Exit Sub
    Else
        If dgvRmServers.IsCurrentCellDirty Then
            _bs.EndEdit()
        End If
    End If
End Sub

Finally, the function that writes my changes back to the database.

Public Function SqliteUpdateKvp(ByVal tbl As String) As Boolean
    Try
        Using con As New SQLiteConnection(_conn)
            con.Open()
            Using da As New SQLiteDataAdapter("SELECT * FROM hosts", con)
                Using cb As New SQLiteCommandBuilder(da)
                    'da.UpdateCommand = cb.GetUpdateCommand
                    Dim testo = da.Update(_dsxConfigDb.DataSet, tbl)
                End Using
            End Using
            con.Close()
        End Using

        Return True

    Catch ex As Exception
        _dsxLogger.WriteMessage(ex.Message, TraceEventType.Error)
        Return False
    End Try
End Function

Now it is possible to write the changed states of the CheckBoxes correctly into the database without any problems.

Summarized _bs.EndEdit() did the trick.