I think we’ve done most of this before, but you seem to dislike the answers, ichkeria. Or at least you seem to keep posing the same questions perhaps hoping you’ll get different answers. I was hoping that by now you would have done a bit of research to establish what is fiction and what is fact. What you need to do is take a glance at this:
http://www.ohchr.org/EN/ProfessionalInterest/Pages/StatusOfRefugees.aspx
This is the UN’s “Convention relating to the Status of Refugees”. The EU's Dublin Convention largely reinforces the Convention but in any case it is the UN document which governs the sort of things currently happening in Europe. Article 31 is the relevant passage which will help us with your misunderstandings. So
“Nor is there any proper definition of what constitutes an illegal immigrant”
That is quite straightforward. An illegal immigrant – certainly in the UK and almost certainly in most other countries - is somebody who enters a country without leave to do so. Countries have varying ways of providing “leave”. The UK, for example, allows people from many nations to enter upon presentation of a passport on arrival. Entrants from some other nations require a visa to be obtained in advance. It is an offence under UK law to enter the country without leave. However, Article 31 provides an exception:
“1. The Contracting States shall not impose penalties, on account of their illegal entry or presence, on refugees who, coming directly from a territory where their life or freedom was threatened in the sense of article 1, enter or are present in their territory without authorization, provided they present themselves without delay to the authorities and show good cause for their illegal entry or presence.”
But note (as I have constantly pointed out) “…coming directly from a territory where their life or freedom was threatened in the sense of article 1, enter or are present in their territory without authorization, provided they present themselves without delay to the authorities and show good cause for their illegal entry or presence.”
This also addresses your constant contention that they are entitled to roam from country to country. (most recently today: “ They don’t need to register in Hungary. That is bureaucratic nonsense”).
The UN Convention is quite clear. They must come directly from a state where they are under threat and they must present themselves in the first safe country they arrive. If both these conditions are not fulfilled the exception provided by Article 31 is void. You may call it nonsense. That is your privilege. But it explains why almost all the migrants arriving in the EU are illegal entrants because almost without exception they will have previously passed through countries where they are not under threat. I hope this helps put this misunderstanding to bed.