From what I understand, the sell-side quants deal mostly with PDE, FDM and numerical methods like Monte Carlo sampling etc. , and the buy side quants deal more with statistical modelling, prediction, Machine Learning and statistics. Am I correct in making this assumption.

It doesn't help that there are many quant titles such as Risk, Front Office, Pricing, Fixed Income, Equity, Algorithmic Trader etc. I'm very confused as to which position I should apply for. I want to get my foot in the door as a quant.

Could someone please help point out which Quants require what skills in mathematics and if it's easy to transition between these positions once you're in the door.