Horses can swim, and some like swimming in the ocean. However, just like humans walking in the sand is tiring for horses, and swimming in an area with waves or tidal pull is also tiring. A horse should not swim in the ocean for more than 10 minutes or so. You mentioned that the pictures show the water coming up to the riders waist. That means that the horse is completely swimming. Any person riding like this should be an excellent swimmer, and I would not recommend this with a horse that is new to you. To make sure the animals are well cared for, you should ask ahead of time if the horses are immediately washed with freshwater after going into the ocean. This helps to protect their coats, particularly where the saddle and tack touch them. Just like humans, salt water and beach sand can irritate a horse's skin.