You need planning permission, because it is two storey. You will need Building Control/Structural Engineer to confirm that your existing foundations are suitable for a two storey extension.

Any second floor extension will need planning permission.
The 4.000m height refers to single storey extensions.
Loft conversions do not need planning permission if you say used Vellux type windows however with a dormer window planning approval would be required.

The 4 metres threshold refers to the highest point that can be reached by the roof of the ground floor extension, in case of a sloped roof; the reason for this limit is easy to understand: it prevents the roof of the ground floor extension from encroaching with the windows of the first floor.

With regards to the upper floor extension, it DOES require a planning permission; whilst it is still possible to obtain planning consent for 1st floor extensions, these type of extension should be carefully designed in order to avoid a) overdevelopment b) loss of daylight to neighbouring properties, which are usually the grounds for refusal for this kind applications.

Hi
You mention the house is a semi detached house. An additional storey on top of the ground floor extension can only extend out from the original house wall by a maximum of 3 meters under permitted development legislation.
A loft conversion with a dormer is acceptable as long as the total volume created by the new room does not exceed 50 cubic meters. If the extension is within 2 meters of a side boundary then the extension can not be higher than 3 meters.
Hope this helps.