Caution is the watchword indeed but don't let it become a thing or it will spoil your walking. The odds of being injured really are low but certainly not zero. Every encounter is different. I always watch beasts in fields like a hawk and try to read them. Some need a very wide berth to the point of using dead ground if necessary, some can be approached freely; their bearing should tell you what's right but always keep your options open.
I have felt the sharp end of a cow. I was taking a photo on a col with lots of placid cows that were well used to humans, too used as it turned out. I had my back to one which was a couple of feet behind me when it decided I was on its lunch and nudged me out of the way with its horn - right up my backside. I got the point and moved sharply but what really hurt was Mrs N laughing like a drain.
The only farm animal that has pressed home an attack on me was a cockerel who decided I was too close to his ladies and launched an attack. I never saw it coming, the first thing I knew was a flutter and a stinging pain in the back of my thigh. I have also been attacked by buzzards and gulls
My personal ranking of animal hazard in descending order of risk would be motorists on narrow lanes, birds (especially geese), dogs not under control (especially farm dogs), cows, horses and mountain bikers. Above these in terms of injury risk, I would put loose and slippery rocks, greasy surfaces and barbed wire.