Theine is a historical name associated with caffeine, not a separate special form of stimulation unique to tea. Authoritative chemical references, including NIST and FDA substance listings, include theine among caffeine's names. That resolves a naming question without making every caffeinated drink identical in preparation or concentration.
Theophylline and theobromine are related chemical names but should not be casually substituted for caffeine. The source moves from terminology into pharmacology and treatment claims; those passages are not a sound basis for using tea or coffee as medicine.
For beverage comparisons, identify the product, serving size and preparation instead of deciding from a historical name alone. If the question concerns a health condition, medicine or a reaction to caffeine, use qualified medical advice and current product information. A useful coffee glossary can clarify language while leaving individual treatment and dosing decisions where they belong.
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A good question can make a better brew.