How to specify tints for white wallpaint?
You select paint one of two ways. 

Use hardware store paint chips

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You go to any hardware store and grab from their array of paint chips, the vast majority being pastel off-whites just like you are looking for, because everyone who paints their house wants the same thing as you.  Then you fiddle and fidget and maybe get some $6 samples and do some test patches or whatever, until you find the color you are happy with.  

Then you go back to the hardware store, and get some more of those paint chips of that same color, draw X with a sharpie on the colors you do not want (they come 3-6 to a card), and hand that to your paint supplier and say "MATCH!!!" 

It's not necessary for it to be the same hardware store or paint line amongst the three: Paint chip vendor, sample vendor, and final paint vendor. Though it does work better if the paint chip supplier matches the sample provider, once you have the color dialed in,  you can send the paint off for match to anybody. 

Or, use paint chips of your own provenance

Look at any paint color system like Pantone, CIE, Munsell, RAL, or whatever source suits you.   Again, pick the paint chips that  you like.

You can also take that chip in to have any competent hardware store scan it and match it, however they may not be able to give half-pint sampler cans; you may have to get a whole quart.  

You can also custom-mix sample sizes of paint from whatever compatible paints you have lying around. When you're near-white, you can start with snow white and add a very small amount of a deep color as needed to tune it where you want it.  Once you have a few tablespoons of mixed paint you're happy with, paint it onto some drift cards that have been pre-painted a very close primer color (i.e. white). Paint 2 coats.  Now you have "drift cards" you can present to any paint supplier and say "MATCH!!!"  Keep more of your drift cards on file, to make sure the match is true. 

For ordinary paint suppliers, the science of matching colors is well-established, and they have no excuse not to succeed. 

The only case where someone might have trouble is in dealing with narrow consumer-tier systems like Rustoleum, where they only sell a couple dozen colors (but controls are very, very good between cans, which you can't say with a custom mix).  In that case you might have to say something like "10 parts J123 cloud white and 1 part J234 barn red" or whatever.  The industrial-tier suppliers I deal with will cheerfully develop custom paint formulas to match your drift card.