We consider an energy harvesting information update system where a sensor is allowed to choose a transmission mode for each transmission, where each mode consists of a transmission power-error pair. We also incorporate the battery phenomenon called battery recovery effect where a battery replenishes the deliverable energy if kept idle after discharge. For an energy-limited age of information (AoI) system, this phenomenon gives rise to the interesting trade-off of recovering energy after transmissions, at the cost of increased AoI. Considering two metrics, namely peak-age hitting probability and average age as the worst-case and average performance indicators, respectively, we propose a framework that formulates the optimal transmission scheme selection problem as a Markov Decision Process (MDP). We show that the gains obtained by considering both battery dynamics and adjustable transmission power together are much higher than the sum gain achieved if they are considered separately. We also propose a simple methodology to optimize the system performance taking into account worst-case and average performances jointly.