No, you don't need any DPM.
Both floors you have have got laminate foam underlay as
@Andystiletiling said. This can just be thrown away. I was also right (in a roundabout way!) in that you have old Marley thermoplastic tiles which are stuck down with bitumen (black) adhesive. These are brittle and if they are breaking up near the perimeter then it is a pretty good sign most others will soon follow suit. They usually contain very low grade asbestos but check with your local waste disposal site to check that they accept these, most do in local landfill. Take care when removing for dust etc and most usually come up whole or near whole.
You don't need hardiebacker from what I have seen, but could benefit from slurry coating over the whole floor where bitumen adhesive is present and tiling that.