Yes, basically what you have is partially "restored" teak. By the end of your process, before oiling, it should all be a "natural" color. What you need to bring it to that point depends on how weathered and worn/stained it is. With teak, to preserve it's service
life, you want to start with the least destructive approach first, see if that gets the results you want, and move on to a more aggressive treatment if it does not. Casey's prescription is a basic teak cleaner that cleans dirt and stains off the teak and restores lightly weathered wood. The next step is a "one part" cleaner, which is a weak solution of oxalic acid. All one part teak cleaners are basically composed of this. If you have acres of teak, do yourself a favor and buy a jar of the crystals that you mix into solution yourself instead of getting raped buying
the pre-mixed quarts of it at West Marine
If you had some joy with your most recent approach, doing it again may give you incrementally better results. The key is to let the solution "soak" on the teak for 5-10 minutes. If that does not yield an even color, move on to the one part teak cleaner. And if that does not work, you need a two part cleaner, which is more destructive to the wood.
You want to scrub across the grain because the pulp between the hardwood in teak is relatively soft and easily worn away by the bristles in the brush if you scrub with the grain. Teak is very hard overall because silica in the soil of the tree is embedded in the hardwood of the wood as it grows, making it very resistant to wear, IF you don't scrub out the soft pulp.
The best approach is to use a Scotch-brite scrubby pad and run it perpendicular or diagonal to the grain, no matter what solution you are using. Use only as much pressure as you need to achieve the results you're looking for.
I've not used the Starbrite Tropical Teak Oil but have heard very good things about it, compared to other teak oils used on exterior teak. But, as with all brightwork, no matter what you coat it with, the key is conscientious annual maintenance
. Let it slip for a year and you'll quickly find yourself back at square one.