As consumers' green awareness continues to grow, the level of a product's eco-friendliness and the quality of its marketing have become significant factors in shaping consumers' purchasing decisions. The power structures within the supply chain, as well as corresponding government subsidy policies, are also key elements influencing sustainable strategies for the green supply chain. In a green supply chain comprising one manufacturer and one e-commerce platform, two sales models exist within the online e-commerce platforms: reselling and agency selling. This paper establishes and analyzes three distinct Stackelberg game models, namely: manufacturer-led model without subsidy (bm), manufacturer-led model with subsidy (sm), and platform-led model with subsidy (sp). The results are shown as follows, with the rise of consumers' environmental awareness, more consumers opt for green products, inspiring the manufacturer to increase its optimal greenness and platform to enhance its optimal marketing efforts level. Notably, government subsidies provide a significant stimulus. An increase in the green technology cost coefficient leads to a decline in the manufacturer's profits across all three modes. Intriguingly, the manufacturer's profits are always highest in the sp mode. As the marketing efforts cost coefficient increases, the platform's profits decrease in the bm and sm modes. In contrast, in the sp mode, the platform's profits increase rather than a decrease. The choice of mode primarily depends on the platform's marketing efforts cost coefficient. When this coefficient exceeds a threshold, the platform chooses the sp mode. However, due to the relatively low marketing efficiency in this scenario, the manufacturer prefers the sm mode. For the government, the sp mode involves agency selling, serves as an effective mechanism to redistribute subsidies, thereby yielding the maximum social welfare benefits. Management insights are provided for the manufacturer and platform managers to make decisions about the degree of greenness and marketing efforts level, along with insights for governments to optimize subsidy policies.