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The "main" Colorado River starts in Colorado, goes southwest in that state and Utah, and heads west then south through Arizona before forming the California-Arizona border, and empties in the Gulf of California. I would just call it the Colorado River.
The river you are talking about was named "Colorado" because it is "colored," but starts and ends in Texas, and I would call it the Texas Colorado River.
"Colored" in this context means the color of the soil, typically red or brown, and not non-soil colors such blue, purple, or green.