I grew up with horses and have had them for all of my life. Our horses were working animals and we would use them from May until October every year. We used saddle horses and pack horses to take hunters on trips deep into the mountains for weeks on end. From what I have seen I think that they do have distinct personalities and are capable of rational thought.
Here is just a brief rundown on some of the more memorable horses.
Jughead, was an old swaybacked pack horse that us kids (under 5 years old) used to ride. We would pour a bucket of oats on the ground for him because we knew he couldn't resist them. When he lowered his head down to eat them we would swing a leg over his head then shimmy up his neck til we got to his back, then turn around and wait for him to stop eating. Once he had stopped we would use our little legs to kick him in the ribs and he would then go into the corral, trot around the corral for a few laps then head over to the cook shack and stick his head in the door until Mom would take us off is back. We would do this for hours on end until we learned how to put a halter on the horse and make him go where we wanted.
Little blue was a short blue roan that had a lot of heart. When we got older we would ride her all over the country on our various adventures. We also used her for the Gymkhana events at the local rodeo. At one rodeo my sister used Little Blue for the same events. I was 8 and she was 6. When I rode the horse over the hurdles she raced at a full gallop and jumped over all the hurdles like she wanted to win. I took third in this event. When my little sister used the same horse for the event, Little Blue trotted up to the hurdles, stopped, carefully stepped over the hurdle and then trotted up to the next one. My little sister was kicking the horse in the ribs with her legs and whipping it with her bridle rains but the horse would not go fast with her on its back. My sister took last place but got a prize anyways because the crowd was laughing in hysterics. We have that on 8 mm film and I would love to get it digitized.
Napoleon was an Arabian cross that my dad bought off one of our Indian Guides. This horse loved chasing animals. When we were rounding up the horses all we had to do was get on his back and hold on for the ride. Seriously, we seldom had to use the reins when moving horses because he did all the work. One time a friend and I were out for a drive and saw that our horses had got out of the field. We had no ropes in my friends truck but I walked up to Napoleon and got on his back to see what would happen. That horse chased all the other horses back to the field and through the break in the fence for me by himself. He also had a bit of quirk to him. It seemed that he could sense if you were becoming unbalanced if you were running him while bareback. When this happened the little bugger would stretch out his front feet and lock them up, and lower his head to come to a short sudden stop. More often than not he would lose his rider doing this and when he did he would always turn to face us and I swear that critter would be smiling at us.
Brown Sugar was a saddle horse. Period. If you put a saddle on her back you would never have a problem riding her or using her. It didn't matter how skilled the rider was, she was just a good horse. But if we put a pack on her back you would have nothing but trouble. She would try to rub the pack off on a tree, buck it off or even break it by trying to roll it off. She just didn't like packs. (Its too bad one of our new guides didn't believe us and thought it would be a good idea to wrap six dozen eggs up in his bedroll, then packed it on her.... but that's another story.)
Tequila was the horse I loved to ride when I was guiding. That horse would go anywhere I pointed it without hesitation. It seemed that she liked going hunting with us. But she did have her own little quirk. Every time I saddled her up in the morning she would buck for about 4-10 seconds. If she managed to throw me I would just give up and saddle another horse because that roan would fight me for the rest of the day.
Jewel was another great horse. She was calm and great with kids or nervous riders and never gave us any problems. One morning we were deep in the mountains and when I went to wrangle the horses in the morning I found out that they had crossed the river and I could see them on the other side. It was early October and it had snowed the night before and the river came straight out of a glacier so it was cold. I found a shallow crossing where the water was only up to my hips and grabbed a long stick to use as a brace to help me fight the current as I made my way across. My stick slipped and the current pushed me over and carried me downstream until I finally managed to swim out on the side the horses were on. When I finally got to the horses they looked at me and all took off at a run back across the river towards camp. At this point I was beat. I was cold, wet and had the energy sapped out of me from swimming across the river. My lighter was wet so I couldn't even build a fire to warm up with so I was basically screwed. On this trip it was only my hunter and myself. Axel was from Austria, and this was his first trip in the mountains. He could barely speak English so our communications were mostly signing and pointing to things. He had no idea how to saddle a horse and even if he found me I couldn't tell him what to do. As I watched the horses all head across the river I just stood there lost, because I didn't know how I was going to get out of this jam. Then old Jewel stopped, turned around and looked at me. She was already a third of the way across the river with water up to her knees, then she turned around and came back to shore so I could get on her and rider her across the river. I can tell you this, that horse was pampered for the rest of her life after that day.
I could go on for hours with tales of our horses but to get back to your question; Yes I think that animals do have distinct personalities.