Does refraining from visiting relatives in their homes, and merely greeting them in public places, visiting their sick, and attending their occasions, constitute a severance of kinship ties if it is due to the mother's anger at their actions?

As long as you maintain ties with your paternal uncles and aunts by greeting them, attending their celebrations, and visiting their sick, then you are not severing kinship ties. Connection is achieved through whatever is conventionally considered a connection, such as giving money, fulfilling needs, visiting, writing letters, exchanging greetings, and similar acts. It is advisable to strive diligently to reconcile between your mother and your father's family, as there is a great reward in it, based on the saying of the Prophet, peace and blessings be upon him: "Shall I not tell you what is better than the degree of fasting, prayer, and charity?" They said, "Yes, O Messenger of Allah." He said, "Reconciling disputes, for indeed, discord among people is the razor (that shaves away religion)."