To nudge is much more subtle than provoke. It can be done in a neutral fashion (to elicit any response) or in a way that is directed towards a specific response, behavior or outcome.
nudge (v.)
To encourage or persuade someone to do something in a way that is
gentle rather than forceful or direct:
He cleverly nudges us toward the conclusion he himself has reached.
Cambridge
Transitive. figurative. To give a reminder to; to coax, encourage, or steer gently or by degrees in a desired direction. Also
occasionally intransitive with at.
He talked at length to his father who tried to nudge him into the
ministry, but as yet he was too undecided to plump for anything so
definite.
1987 W. Raeper, George MacDonald v. 55
[OED online]
Nudging is a practice that uses tweaks in the way choices are
presented to people to steer behavior, such as changing decision
defaults, asking people to precommit to an action, or making choices
more salient.
Alain Samson; "On Your Best Behavior: How to Nudge
Ethically" at behavioraleconomics.com
In our experiences with quick images, asking students, "How many dots
did you see?" can nudge them to use a counting-all strategy
because they interpret the question as hinting toward a strategy
(i.e., "how many" means I should count).
Antonia Cameron at al,;
Early Childhood Math Routines (2023)
Oliver (2013) considers nudging to be a nuanced approach to
promoting of healthy eating, aiming to elicit voluntary changes
towards desired behavior withing the given institutional conditions of
choice.
Radek Kovács and František Ochrana; Nudging towards
Health (2023)
We know that learners are lazy and leave things to the last minute so
why not nudge them into correcting that behaviour.
Donald Clark ;
Artificial Intelligence for Learning (2020)
Even in areas like anti-smoking campaigns, nudging, while
effective, has worked alongside high taxation and banning smoking in
the workplace and public buildings and spaces.
Keith Dowding; It's
the Government, Stupid (2020)