After fiddling with it, I managed to fix it, although I don't really understand why.
Apparently in Umbraco 6.1.6 you have to add umbraco_toolbar_id manually, so I added these 2 lines
control.ClientIDMode = ClientIDMode.Static;
control.config.Add("umbraco_toolbar_id", "umbTinymceMenu_" + control.ClientID);
to this function, which becomes
public static void PutTinyMce(ref TinyMCE control, string controlId, PlaceHolder placeHolder, UmbracoPanel panel)
{
DataTypeDefinition d = DataTypeDefinition.GetDataTypeDefinition(Static.DTD_ID_SMALL_RTE); // Custom TinyMCE DataType
control = (umbraco.editorControls.tinyMCE3.TinyMCE)d.DataType.DataEditor;
control.ID = controlId;
control.ClientIDMode = ClientIDMode.Static;
control.config.Add("umbraco_toolbar_id", "umbTinymceMenu_" + control.ClientID);
placeHolder.Controls.Add(control);
panel.Menu.NewElement("div", "umbTinymceMenu_" + control.ClientID, "tinymceMenuBar", 0); // Add TinyMCE controls to menu bar
}
That fixed it. My TinyMCE appears again :)