Sometimes yes, sometimes no.
In fact, horses that really enjoy their work can sometimes show a little bit too much enthusiasm for it. But a horse that enjoys its job will often try their best to do everything right.
However, there can be other motivations too. I used to know a little mare, a lesson horse, who would balk dramatically every time she was asked to do something on her own. She'd plant her hooves and go "No, MAKE me"
But once you got her out, she would do whatever she was asked perfectly. Why? Because she was older, smart, and knew that if the person was going to make her do it, then the best thing to do was get it right first time so she could go back to her buddies where she wanted to be. A well-trained lazy horse can also be very well behaved because they know that if they do it wrong, they'll have to do it again. (Which is how we train them to do it right - make them repeat it until they get it, just like a kid doing math problems).