That picture of the Polar Bear zinnia looks nicely white. There are several strains of zinnias that do have good whites. You mentioned "light green" and Burpee's White Wedding zinnias have been accused of that. Perhaps wrongly. The center of White Wedding is green, but the petals seem pretty pristine to me. However, White Wedding is a low-growing compact zinnia, which is not "my cup of tea" because I like to cross-pollinate my zinnias, and stooping over to do it is extra effort that I prefer to avoid.

Polar Bear is an heirloom zinnia breed. The "dahlia flowered" zinnias were introduced by Bodger Seeds in the 1919 to 1935 time period. There were a total of 18 separate colors, and your Polar Bear White was one of them. The problem with any heirloom variety is that you are at the mercy of the seed growers who have been maintaining the purity or lack of purity of the strain. White zinnias are grown in a separate field, to prevent bees from introducing non-white zinnia pollen into the field. That happens occasionally anyway, but the seed growers usually hire some field workers to go into the field at the time the zinnias are coming into bloom and kill any non-white zinnias.

As someone who grew a white cactus flowered zinnia strain next year, I can attest to the fact that those workers who "rogue" the field just killed the non-white zinnias (or almost all of them). They do not do anything to maintain the quality of the plants, or the quality of the flowers themselves. As a result, only about 1% of my white cactus flowered zinnias were indeed cactus flowered. There were many single or nearly single daisy looking zinnias, many of the blooms were much smaller than cactus flowered zinnias (which usually range from 4 to 5 1/2 inches in diameter.) Essentially none of them resembled the picture of a white cactus zinnia. It is quite possible that you will have a similar experience with Zinnia Polar Bear.

I am trying to breed some good white cactus zinnias, so I will be planting some more of those seeds again this year, looking for a few good specimens that I can intercross and save seeds from. I will also plant a bunch of Burpeeana Giants zinnias, looking for the occasional white specimen that I could use as a breeder.

You might get lucky with Polar Bear, but you might want to take a look at Benary's Giant White and the dahlia flowered white zinnia seeds that are sold to market gardeners who grow zinnias commercially for sale to florists. Good luck. That is what we both are depending on.

The Benary's Giants seem like your best bet. The description of that dahlia flowered white hints that they aren't so good.

ZM
(not associated with any product or vendor mentioned or linked)