In my experience the best thing you can do is NOT be rude. A stylist who is going to try to do what they want will try that regardless of how many products you bring and how firm you are.
If they are open to treating your hair according to your wishes, it won't take being rude to them to find that out.
I think the MOST important thing a person can do is WATCH the stylist they want to use. I stood in the doorway of a salon watching the stylist I'm thinking about going to for 20 minutes in the doorway of the salon. People looked at me like I was nuts. Then when I went to speak to her and unfurled my hair
they were staring some more. If you're satisfied with her work after watching her/him and spaking to them, then go in for a conditioning treatment and a rollerset. Too many times we walk into a salon with a stylist we have NO knowledge of to get a touch up. That's CRAZY! Don't DO it! Feel her out SEVERAL times before the scissors or chemicals are even discussed.
As you go to them, TELL them you have products you really loved and wanted to bring to the salon. Let them know you are willing to pay the same amount (this is where most people run into trouble - you bring your own stuff so you think you're entitled to a discount. While that may technically make sense, it's not reasonable. If they all did this they wouldn't make ANY money and while I think they should make some concessions I think taking money off goes too far in the direction of unrealistic-ness on the part of the patron. But anyhow...)...
if the stylist seems to be open to it, great! If not, work on her a little. Say you experienced a lot of progress with these products, are trying to gain length and preserve health and would like for the two of you to work together as a team to accomplish that goal. Make them feel a part of it. But if they aren't open at all, you'll certainly know right away and you can move on.
I request every service, in detail, using correct terminolgy, with a smile and limit major maintenance like trims and chemical services to the 4th or 5th visit.
Be firm, but be nice, and be SMART about getting what you want. It's not all about twisting their arm - it's about getting them to work WITH you to achieve your haircare goals.
It's the in between times that you have to be more firm. Once you're comfotable and they think you're "cool" enough t hat they can just do what they want. There ARE people like that, after all. They arrive at a salon a nd they rely on the stylist to tell them what they need. They may resist at first, but eventually they cave inand do what the stylist suggests. Sometimes that's good and sometimes it's bad, but always it's under YOUR control to some extent - so don't give up your stake in the outcome! Be vigilant!
Make it very clear to your stylist what you want and then watch her, let it be known you're watching her. Ask her to hand you a mirror while she touches you up or trims you. If you think the relaxer is too far overlapped - tell her and ask her to rinse the chemical from your hair. if you don't want that trim - BE FIRM!
Be PROACTIVE, but be NICE.
This is my policy and I can count the number of terrible stylist experiences I've had in my LIFE, on a hand...