Which is higher, the COP or the EER, for the same heat pump? Please ask an expert to explain

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The textbook states that the COP (Coefficient of Performance) is the ratio of the cooling/heating capacity obtained to the energy input. During cooling: COPR = QR/W (R is a subscript, which cannot be displayed here). During heating: COPH = QH/W, where QH = QR + W; therefore, COPH = 1 + COPR. In other words, the heating coefficient of a heat pump is always greater than its cooling coefficient, and it is a value greater than 1. When selecting a heat pump unit in practice, engineering thermodynamics textbooks state that the same heat pump has a high heating efficiency, but in actual selection, the parameters provided by manufacturers refer to its cooling efficiency. The explanation given in the textbooks is that the compression process in the compressor generates heat, and this heat must also be dissipated through the condenser; hence, the amount of heat dissipated by the condenser is greater than the amount of heat removed by the evaporator. Therefore, the heating efficiency is slightly higher than the cooling efficiency. The reason for the high refrigeration efficiency is as follows: based on the parameters shown in the figure above, (1) during cooling, the inlet and outlet temperatures of the cold water are 12/5°C, while those of the cooling water are 24/30°C; the temperature difference of the fluid inside the evaporator and condenser is around 20°C. (2) During heating mode, the inlet and outlet temperatures of the heat source water are 8/3°C, while the inlet and outlet temperatures of the hot water are 38/45°C; the temperature difference of the fluid inside the evaporator and condenser reaches 30–40°C. The greater the temperature difference, the more effort is required to achieve energy transfer; from this perspective, it makes sense that the heating efficiency is indeed lower than the cooling efficiency. These two explanations have puzzled me for a long time! I would also appreciate it if an expert could explain whether a heat pump has a higher COP or a higher EER