In some States of the US they used to have similar laws regarding Negro swimming pools - Negros were allowed to construct swimming pools and the like, as long as they posted signs informing people they were Negro swimming pools, to prevent Negro-Caucasian strife.
Yes, however Ahmadiyyas are not banned from eating, drinking, or even living in the same areas as other religious groups (Muslims, Christians, Sikh, Hindus). Rather laughable comparison.
This specific case was ruled so as to preserve the peace. As the Western society is mostly secular, this concept is understandably foreign. The issue is resolved in the US by property laws, for example Jews can't enter Baptist churches to preach their religion and secretly convert churchgoers, because the owners of the property would have them kicked out.
Pakistan's laws are designed to prevent subversive preaching, under a false cover, which all Muslims (Sunni/Shia/etc) jointly petitioned the government to put an end to. They can still preach their religion and practice their religion, but only under the guise of their religion, not some else's religion.
As far as the definition of a Muslim, one has to believe the following:
1. There is one and only one God, the God of Abraham
2. Prophet Muhammad (peace be to him) is the servant and final messenger of God
There are further six articles of faith, belief in God, his books (revealed Torah, Injeel, Zabur, and the Quran), His angels, His prophets (Abraham, Moses, Jesus, Muhammad being the major ones (peace be upon them all)), the Day of Judgment (Afterlife), and Al-Qadr (supremacy of God's will, knowledge of all things at all times).
Islamic scholars have reached a consensus as to which groups are Muslims, and which groups are not. This was due to the rise of heretic movements.
Ahlus Sunnah wa al Jama'ah (Sunni) = Four Madhabs of Imams Abu Hanifa, Shafi'i, Malik, and Ahmad bin Hanbal. The Salafi, Wahhabi, and Ahle Hadith.
Shi'a/ Shi'i = Madhab of Imam Jafar (Ithna Ash'ari = Twelver), Ibadi
The Sufi are not technically another sect. All groups mentioned above, except the Salafi, Wahhabi, and Ahle Hadith are Sufi (in the Western definition). Furthermore, Sufi is a genetic word for teacher or Imam in many regions, and thus a great teacher and holy man is referred to as a Sufi. So those who believe Sufism (personal spiritual insight through contemplation, zikr), are not Sufis until they become teachers themselves. Much of it is based on spiritual lineage as well.