The question seems to be essentially:
What does "under control" mean?
Does it mean the character is braindead while not receiving orders? Can he disobey any orders? While not receiving orders, what can he do if he's not braindead?
First Angle: Spells
Let's check what spells give you control over other creatures and how they behave.
Animate Dead gives you a zombie
If you issue no commands, the creature only defends itself against hostile creatures. Once given an order, the creature continues to follow it until its task is complete.
The same goes for Animate Objects, Conjure Elementals, etc...
So, for the situations where you "create" a creature from what was an inanimate object before (or a corpse), they are braindead. Note that the procedure to create a Vampire Spawn is similar to that of a Zombie or a Skeleton. So far, if you take this angle, your Vampire Spawn is braindead.
Second Angle: Attributes
But hey. Zombies, skeletons and elementals have the Int attribute of an animal, so yeah, they are braindead by default. Our Vampire Spawn has an 11!
So, how do characters with intelligence behave when being controlled?
Let's check Dominate Person
If the creature completes the order and doesn't receive further direction from you, it defends and preserves itself to the best of its ability.
Well, that didn't help.
Third Angle: The Role-Play and the Game
Well, all these spells were intended for you to create a temporary combat minion. They don't actually have personalities, backgrounds or whatever. Probably that's why they are so blunt, not doing anything besides trying to survive. You can have Vampire Spawns that are just like that, combat minions to your Vampire Lord and that's all. But you seem to be interested in something else.
General rules I think everyone could agree on what being "under control" means:
- If given an order, the character will try to accomplish it to the best of its capabilities, preferably as intended by the ruler (meaning no wording play, unless actually misunderstood).
- The character can not try to do any kind of harmful action against its master.
I think these two basic rules are enough for you to role-play an "under control" character.
Specifically for the Vampire Spawn case, it explicitly says it has no free will (as he only gets free will when the creator dies), so an additional point would be
- They lack any kind of motivation to do anything besides surviving.
This last point is, IMHO, the one you can tamper about if you want to make a player have a Vampire Spawn character. They might have memories of their past lives and still have some feelings, motivations and maybe even morals, so, unless it breaks the first two rules, they might be able to do something other than sucking blood to survive and sleeping.
The rules intend any kind of minion (Zombie, Vampire Spawn, Animated Objects, Conjured Elementals, Dominated Person or Monster) to have no free will and do absolutely nothing besides defending themselves and surviving. In the Third Angle, I try to suggest a more broad definition of "under control" than this completely-braindead-character. My suggestion seems to be consistent with the vampires in Curse of Strahd, which were mentioned by the OP in some comments.