The Greatest Question: Quant Finance or Finance?

I am a first-year MSc in Finance student at Bocconi University with a Bachelor's Degree in Statistics and Computer Science from a pretty unknown university in Southern Italy with a 4.0/4.0 GPA. I understand your thoughts: what is a STEM guy pursuing an MSc in Finance at a business school? I'm wondering that too!
Before my admission to the master's in Bocconi, I sent several applications to universities like Oxford, Imperial College, UCL and École Polytechnique to attend the Masters in Quantitative Finance and got admitted to the last two, but the offers arrived too late.

I am enjoying my time at Bocconi; the master's program is engaging, and there are strong quantitative courses. However, I am seriously considering transferring to another university to apply for quantitative finance programs. My goal is to pursue a PhD in Quantitative Finance, and I believe that with a Master's in Finance, I might not have a sufficient quantitative foundation, even though I am quite skilled in that area. 

Given this challenging situation, I am being troubled by two questions:
1) Should I apply for an MSc in Quant Finance or stay here at Bocconi?
2) Which masters should I apply for? I'm searching only for a Europe-based master's, preferably lasting one year and in a prestigious university also famous in the USA. I'm considering only 3 alternatives: Imperial College, UCL and ETH.