Universities are organizational structures with individual activity mixes or strategies that lead to different performance levels by mission. Evaluation techniques based on performance indicators or rankings risk rewarding just a specific type of university and undermining university diversification: they usually introduce homogenizing pressures and risk displacing university objectives--neglecting their socio-economic contribution and focusing on succeeding on the evaluation system. In this study, we propose an alternative evaluation method that overcomes these limitations. We produce a multidimensional descriptive classification of universities into typologies, while analysing the relation between their institutional factors (characteristics) and their (technical) efficiency performance from a descriptive perspective. To do so we apply bootstrap data envelopment analysis (DEA) and multidimensional scaling analysis (MDS), performing a so-called DEA-MDS analysis on data on the Spanish university system, and unlike previous studies, we include data on an important dimension of the third mission of universities (specifically knowledge transfer, KT) in their characterization. We identify six types of (homogeneous) universities. Results indicate that to be fairly efficient, universities may focus on teaching, KT, or overall efficiency but always have to fairly perform in research. Additionally, results confirm the relevance of the third mission as a source of institutional diversity in higher education. This approach could be used to address an alternative evaluation methodology for higher education institutions with formative purposes, evaluating universities according to their unique characteristics for the improvement of HE systems.