Would a more constructive way to describe this be: 'helping people to grow'? To me 'influencing' sounds covert and without consent. They would also personally benefit from using the strategies you'd like them to employ, right? Why try to 'influence' them, when it could be something achieved with their full participation and consent? Then it becomes something you've done together, instead of you having successfully 'influenced' them (read: manipulated).
When talking about this I do acknowledge the constraints of modern day workplaces, and I think with that comes the importance of showing them the organizational constraints - the challenges you face as a leader or manager, in your role - and being open with them about how their various efforts can support you in that role.
"My employee" if you're a manager they're not your employee; "a member of my team" that doesn't make clear if they report to you or you're on the same level of the hierarchy; "my direct report" I just find this a weird/confusing turn of phrase whose purpose seems to be avoiding using the term "subordinate."
E.g. different companies will prioritize different things and the correct behavior is contextual; for a very crude and simplified example, if some employee puts speed over correctness and the company wants them to change their behavior to value correctness over speed (or the other way around! their previous workplace might have worked hard to instill the values you're trying to reverse!) then that's not really 'helping people to grow' but just that - influencing their behavior patterns.
Effectively influencing people's behavior while maintaining trust takes time. If someone is doing 3 things you don't like, pick the one you want to address in the next 2-6 weeks, and let the other ones slide for the time being. If you're building long-term relationships, there will be more time, and as you build trust giving (and receiving) feedback becomes easier for all parties.
Urgency is the enemy in this situation.