Opinion follows. Apply salt to taste. Your mileage may vary.

It's a question of the power dynamic involved in each.

Where the power dynamic is in your favour, you use influence.

Where it is in the other party's favour, you use persuasion.

Where it is equal, you use negotiation.

Imagine a set of scales with the object of moving weight to achieving equilibrium. When you use influence, you are the heavier side and you pass weight to the lighter side. When you use persuasion, you are the lighter side and you draw weight from the heavier side. When you use negotiation, both sides are equal and must exchange equal weight.

Note that although this can refer to differences in status, it doesn't have to. A power dynamic is responsive to circumstances. This is the point behind unions, after all. Get enough workers together and they have the power to negotiate, because they are, collectively, equal in power to their notional superiors.

In reply to Robey:

Thank you, this is the direction I needed.

In reply to Robey:

They should have asked that in the first place Robey.

In reply to David Perry:

Might be worth mentioning that there are typically distinct kinds of power / influence used: eg physical / coercive or charisma / charm / beguilement or financial / fiscal / or by authority / by appointment.

I like to work through Cialdini by relating it to outside of work.

Social Proof - FOMO
Authority - Mum
Reciprocation - Favour
Consistency - 'flakey friend'
Liking - fwends (inbetweeners)
Scarcity - online shopping/DFS

Social Proof - is FOMO (fear of missing out) - it's the reason that we follow trends, it's the reason people can get you to have 'naughty' food when you are being healthy. We don't want to miss out or seen as not following others. Making people think everyone else is doing it, or talking about it. Reason I sat through Stranger things because I didn't want to not be part of the conversation.

Authority - I link this to my 'mum' (you will have one person in your family) regardless of how old I get, if my mother asks me to do something i'm doing it. She has the authority. Authority works by either retaining it and being the 'stern one' or borrowed in which, I got my little sister to do a lot of mundane tasks I didn't want to complete; because 'mum' said so.

Reciprocation - if I do this for you, will you do this for me. Or highlighting 'remember when' so can you do this etc.

Consistency - people struggle the most with this. I explain it as stopping that 'flakey friend' from backing out of doing something. Getting them to confirm exactly what they will do and using it. So you definitely want to go for Mexican food. 100%? at 5 o'clock. Once they've confirmed you can say remember when you said. Consistency sits internal and external if you think you are a 'fun' person - people can use that e.g. 'come on you've got to come out you're always the life and soul of a party' and external is things such as tell me your three most important things you look for in a partner/house - right i've found x,y,z you agreed this is what you are looking for.... will you buy/go on a date

Liking - more likely to do something for someone you like. Extreme push of building rapport, helping others out. 'you know you are my favourite' is more chance of influencing someone.

Scarcity - fear of loss. Remember when DFS was constantly 'closing down' so they had sales, people went mad. My family bought two sofas we didn't need. Or when you spend an obscene amount of money on online shopping because it's free next day delivery.
There's only two spaces left, i know a few people want to go, but i wanted to ask you first.

Hope it helps.
Sorry it's lengthy

In reply to Robey:

Oh my GOD this has been SO helpful!! Thank you!

In reply to Wendy:

In reply to Robey:

Is there an academic reference I could use for the power dynamics you have described?

In reply to Laura:

I'm afraid my definitions are entirely my own work, Laura, and based only on my years of personal, practical experience. Feel free to quote me, though.

Interesting, re-reading my post in the light of the Brexit "negotiations" as each side tries to persuade the other that the power dynamic is, in fact, in their favour...

In reply to Robey:

Hi Robey, I hope you don't mind but i found your comments really helpful and so i am going to quote you for my CIPD level 5....now i just have to work out Harvard referencing, for a thread!

In reply to Robey:

This is great, Robey. How do I reference you?

In reply to Nicola:

Harvard referencing system on how to reference a web source:
"To reference a website in Harvard style, include the name of the author or organization, the year of publication, the title of the page, the URL, and the date on which you accessed the website. Author surname, initial. (Year) Page Title. Available at: URL (Accessed: Day Month Year)."

So:

Jenkins, R. (2020), CIPD Community Forum, Available at: [url] (Accessed: 4 March 2021)