Technically, aren't they kind of both? Given that the UK government have removed all legal avenues for those seeking asylum to get into the UK.

Maybe not all of them, though. Maybe some asylum claims have been going so long (because the government isn't funding a civil service to process them and is funding hotel owners instead) that they arrived here through a now closed legal channel. Similarly, not all illegal migrants apply for asylum, or even commit the crimes that the Daily Mail harps on about.