Financial Mathematics.....well, is just quantitative method (i.e. applied mathematics + statistics) with application in finance (main tools are differential equation, numerical method, stochastic process & calculus, time series and etc...), and thus if one is doing research in this area, seldom they will do it under PhD in Financial Mathematics since the field is not big enough to support a department/field, but there are lots of researcher in various department doing so (e.g. PhD in Mathematics, Statistics, OR, Finance, Economics.......)
If you are interested in the field, but not sure if want a master for a job, or want to continue towards PhD, then the best solution will be to apply for a more mathematical / academics master in quantitative / mathematical finance (i.e. most likely avoid those hosted by business school)
1. generally a master in financial mathematics will cover quite some statistics that are useful in finance (e.g. financial time series, and also various risk measure (EVT), though just some will go into Bayesian estimation or various computational statistics stuffs which also have tons of financial application)
2. generally a financial mathematics master has better placement than a statistics master, not because it is mathematics vs statistics, but because the technical are more focus on one certain application area (i.e. industry).
3. which PhD......well, academics interest is the key factor. ask oneself.