A sheep farmer is a person who farms sheep. This is usually going to involve herding the sheep at some point but it is possible to be a sheep farmer and pay someone else to do the herding, just as it is possible to be a carrot farmer and never actually pick or carry carrots.
The definition of a shepherd is much more restrictive. A shepherd herds sheep. This essentially entails watching the sheep and ensuring that they stay together and are kept safe, rather than being allowed to split up into multiple flocks and wander off.
In normal usage of the word a shepherd doesn’t move sheep anywhere in particular, she just moves them around the grazing territory to give them access to fresh feed and water. Often this is a rotational movement. A person who is moving sheep from point A to point B is normally referred to as a sheep driver/drover rather than a shepherd. These people drive sheep from point A to point B, point B usually being markets, new owners or newly acquired farms.
The living arrangements of shepherds is of course varied. Often they are the farmer/owner of the flock. As such they will be living on the farm. Many will be employed by the owner, and may either live in the nearest town etc or else have accommodation on the farm. In areas like Australia where the farms are huge and the nearest town may be several hours drive away of farm accommodation is rare.
These days shepherds have become almost non-existent in the western world. The development of netting and electric fencing for flock containment and poisons and rifles for predator control has essentially rendered shepherds obsolete. The only time that sheep need constant supervision is when they are lambing since some breed shave real problems giving birth and they are also very vulnerable to predators while giving birth.
In less developed nations and in pre-industrial times shepherds play/playeded a much bigger role. Land was commonly unfenced and sheep needed to be actively kept together. In the middle east and other dry regions human memory of water points was also essential to ensure that sheep could gain access to all available pasture. Without a person to drive the sheep to a distant water point they will only graze within a few miles of known water, returning every evening.
Shepherd is a term reserved exclusively for people who herd sheep. The terms cow-herd, swine-herd, goat-herd (yodeli yodelay), etc are used for other types of herdspeople. I guess that technically a llama herd would herd llama. The term shepherd has been retained in English for a couple of reasons. One of the primary one is its importance in the NT. The other is that sheep require more care than cattle or goats, and can’t be penned like pigs. As such the role of the shepherd remained important for much longer.
There probably isn’t a different definition of shepherd in other nations, but the role for them is more or less important. In Australia the role of the shepherd was never great because of the time of settlement. By the time the sheep industry was in full flight wire fencing and poisons were probably already in existence. I have read accounts of shepherds being killed by Aborigines during the early years of the colony so there were shepherds there at some stage. In sub-Saharan Africa I gather that goats were traditionally favoured so the traditional occupation of Shepherd probably never existed prior to European colonisation. Post colonisation the situation would have been very similar to Australia.