What's a better way to explain motive vs goal?
Want vs Need
What characters want
The character's longterm desire is a want. This is a dream or fantasy, something personal, that is unlikely to happen under their current circumstances when the story opens.
The want is 'pure' because it hasn't been acted on. It doesn't have to be saccharine or lofty, it could be a brutal revenge fantasy, or vapid influencer popularity, or to see Paris someday – it's a character-thing.
In musicals, the want is spelled out in a literal "I Want" song, probably at the earliest quiet moment just after the MC has been disappointed, scolded, reminded they will never go to a ball, or kiss the prettiest girl. It's before the inciting incident.
Audiences get a two-fer to bond with the MC: a show/don't-tell incident where we learn something about the rules in this world and (according to the protagonist) how some are treated unfairly – immediately followed by the MC retreating into their fantasy. Time stands still, the MC gushes their most naive desire, and the audience 'heals' from the bad experience through this false catharsis. Nothing has changed for the MC externally, it's just a coping mechanism. We see how this character is tapping into their want when they are in a dark place.
Rather than feel shame, abuse, fear, misery… they feel hope. This makes them a small hero even though they haven't done anything. It's the setup for this character's journey. Narratively, it might be a naive pantomime of the story's theme, or a twist on this character's eventual arc. The want is often like the monkey's paw curse. What they get is not at all how they imagined, playing into the theme.
What characters need
Contrast the want with what the character physically needs: shelter, food, a home for her family, money to pay the bills, and a support network of friends/family for when something goes wrong.
These needs are physical and practical. Most are not possible without compromise, ie: a job, access to the economics of your world. Without the physical world, the stakes have no meaningful consequences. It would just be characters wanting and desiring things they haven't earned.
Needs are generally worldbuilding and plot (father in a poor-house, mother dying of tuberculosis, and it's about to rain!). Also deus ex melodrama: your protagonist is as unfortunate as possible while maintaining suspension of disbelief, apropos to the tone and genre.
There are psychological 'needs' (affection, self-worth, freedom) but narratively these work like a want that must be earned through change/sacrifice.
Plot-wise, a need is a motive that compromises the character:
She needs the medication for her child, therefore she compromises her integrity to get it. We feel sympathy.
She wants to be a moviestar, therefore she compromises her integrity on the casting couch. We feel unsympathetic. It wasn't earned.
Consider a frog, slowly boiled in a pot
It's easier to figure out a character's motive scene-to-scene, when they are believable people in a semi-realistic world.
To keep a roof over her kid's heads she needs a good job – but who will care for her kids while she's at work? Now she needs a great job and a nanny, but she's out of the career track for years so that's not going to happen. Her husband agreed the mommything would just be a brief pause, except now he's too busy to step in as 'Mr Mom'. She's better at this than he is anyway.
They moved further from her family (her support network) for his job, or that was the pretext since he never liked spending time with her friends/family. Being an abusive jerk, his career isn't so hot but there's plenty of other people to blame when he feels insecure. She's been compromising little-by-little, now it's his turn and frankly he doesn't have to compromise. He's in a unique position to sabotage any situation that doesn't work for him, all he has to do is... nothing.
Your MC doesn't need to be brainwashed or gaslit to end up in an un-equal situation with a miserable person, it's pretty common. Compromise her needs, one-by-one, and you have the pressure-cooker suspense of a thriller (or horror, or melodrama) – definitely an arc you want to play over time, show/don't-tell.
The inciting incident will directly compromise her needs (plot-driven), or alter the status quo of her want (character-driven). As the plot progresses, the other will also be effected (at the lowest point of the story she gives up on the want ~or~ she escapes the abusive husband by living her fantasy) until finally a new status quo ends the story.
Yikes…!
I made the generic suggestions above because there are some red flags in your description of this character, like maybe she was born to serve the plot, lacks her own agency, and might need character development.
She has been brought up to believe that she is weak and easily controlled.
I don't think you can convince someone they are weak and easily controlled.
You can have a local government that officially decrees that women are weak and easily controlled. You can have a religion that indoctrinates it. A corrupt doctor could write it on her certificate of parental fitness. The husband can login to Q-anon pickup artist websites that will definitely confirm women are weak….
I suggest instead that she is isolated from finances and her support network. She possibly has an 'other' status that makes her believe the locals would not help her. In an extreme (but not fictional) world she might be enslaved in a territory where she would be returned if captured. In another time she may have eloped and her family disowned her. Very recently she could not have opened a bank account or rented an apartment without her husband's signature.
Acknowledging there is nowhere to go is maybe more soul-crushing than gaining a plot-imposed hollywood mental illness.
She yearns to teach her children to be strong (like she wishes she was)
This is maybe her want?
Strength is not really a feminine want. It feels like a placeholder until something better comes along. She would teach her kids what she believes, or tries to believe. Where her beliefs radically differ from her husband or the mainstream, she would keep it covert. Kids don't keep secrets.
There is a built-in timer where her children will someday not be dependent on her and her situation will change, but it is years away. She is weighing all her needs against this deadline, possibly countless smaller deadlines that might incrementally regain her some freedom.
Her inciting incident, the reason she grabs the kids and runs, is because she no longer considers 'waiting out the clock' to be an option. Combine the husband's escalating abuse with her children's awareness (or indoctrination). At the same time she can no longer hide in her want fantasy, so she decides to make the fantasy real. As a character she is making choices that we have prior feeling about. It makes us care what happens to her.
but has gotten into a situation where she is fully under the
control of a domineering husband.
Remember that needs are physical and very real. 'Fully under control' means he can sabotage her car, her phone, her mail. If she goes to the police, he will distort the truth. If she is held in custody, she can't prevent him harming the kids.
This has also worked the other way. Her needs are practical reasons she stayed – but having compromised herself she will be judged for it. He supported her while she stayed home with the babies.
Women are 'weak...' suggests you have a medieval or religious dystopia setting, but consider how my real-world examples can translate to your world. Readers will recognize the truth in it.
I think her goal is to free herself and her children from him... all I
can think of is that her motive is to protect her children.
You've described a caged animal, not a woman. She is just mommy-hormone survival instincts defending the cubs.
I can't think of a bigger 'yikes' when describing a woman.