In response to growing global concerns surrounding sedentary lifestyles among adults, the imperative to motivate physical activity has driven the exploration of Virtual Reality (VR) exergames as a promising solution. Personality traits, particularly competitiveness, can play a pivotal role in influencing individuals' motivation to engage in either competitive or non-competitive versions of the same exergame. While existing studies have primarily focused on sports-related competitive orientations, they have overlooked the potential gain in information from gaming-related personality traits. This research addresses this gap by using a modified gaming-related competitiveness questionnaire and evaluating a VR cycling exergame featuring solo, self-competition and virtual opponent cycling conditions in a controlled user study with 38 participants. Empirical findings reveal a notable difference in self-rated competitiveness between the two domains of sports and gaming, and significant correlations between some gamer-types and motivation. These findings highlight the importance of these personality types during exergaming and their potential efficacy for personalisation to increase motivation.