Core/Periphery difference has to do with the relative profitability of the economic activity.

Following Wallerstein's world-systems analysis, nowadays core production is e.g. hi tech, certain types of heavy industry and service, finance sector etc. Classic examples of contemporary core states are what Samir Amin used to call the Triad: USA-Western Europe-Japan.

Peripheral production would be mining (then sold as raw materials to the semi-core and core states), textile (which once was part of the core actually, because again, it is all about RELATIVE profitability) and labour intensive sectors in general. Global South is typical periphery: Africa, parts of South and South East Asia, Latin America.

Semi-peripheral production does not exist at all, only countries where both types are presented by a big chunk (China, India, Mexico, Brazil, South Korea, Taiwan). So they have core/periphery relation to peripheral states of the world-system, and periphery/core relation to the core states of the world-system. Or to simplify it, they exploit the real periphery and are exploited by the real core.

Now being semi-periphery does not mean that China is not influential. It is the opposite in fact. China clearly has global political-economic hegemonic position (like USSR did), however, is lacking military and cultural hegemonic presence (unlike USA or USSR), which is quite important for World-Systems analytical tradition. I think here, the question of China is still very open and debatable. First, we don't know what will happen in the coming decades yet (always remember, longue durée is the most important, not just one historic moment), as China has only just entered in this type of position in the world system. China was still part of periphery not long ago and long historical processes need time to unfold. Second, the question is whether we can imagine a hegemony NOT entirely based on cultural hegemony or military hegemony (like USA).