As the use of LLM chatbots by students and researchers becomes more prevalent, universities are pressed to develop AI strategies. One strategy that many universities pursue is to customize pre-trained LLM as-a-service (LLMaaS). While most studies on LLMaaS chatbots prioritize technical adaptations, we focus on psychological effects of user-salient customizations, such as interface changes. We assume that such customizations influence users' perception of the system and are therefore important in guiding safe and appropriate use. In a field study, we examine how students and employees (N = 526) at a German university perceive and use their institution's customized LLMaaS chatbot compared to ChatGPT. Participants using both systems (n = 116) reported greater trust, higher perceived privacy and less experienced hallucinations with their university's customized LLMaaS chatbot in contrast to ChatGPT. We discuss theoretical implications for research on calibrated trust, and offer guidance on the design and deployment of LLMaaS chatbots.