I do not fully understand the logicality of your post Chetan. You seem to be a bit all over the place with your planning and a touch paranoid.
Firstly, if you were to actually end up going back to India in 5 years, a PhD in Financial Math is an absolute overkill. Most quants in India have a degree in CS/IT with relevant experience. The icing on the cake is that recruiters probably look for is an IIT degree that you already have. Ashok can confirm or deny this just to be sure but I have other friends in Mumbai who'd agree with this.
Secondly, 5 years down the line recruiters are largely going to base their decisions on whether to interview you or not based upon the prestige and exclusivity of the firm(s) you've worked for in the last 5 years and/or how compatible has your experience been over the last 5 years to the job profile they're looking to fill. The reputation of your degree is almost never going to matter. I'm almost certain this is the way in the quant/trading space. Ofcourse, it's entirely different if you were intending to pursue a career in PE/VC/Management Consulting where it is the exact opposite. Your alma mater essentially decides in the latter case as to whether you get an interview or not since they like impressing their clients by all the candor and bravado they can while telling them your a Harvard/Wharton/Stanford MBA !
In the quant space and/or trading, your innovation, sheer mental aptitude (measured via your Math ability), ability to program efficiently (know Data Structures and Algos in and out) and ability to gauge the markets matters only. Nobody cares whether your from MIT or from Pimpri College (although convincing them with a Pimpri College degree would be nothing short of climbing Mt. Everest allegorically)!
If you do decide to do a PhD to become a Quant, I suggest that you do it directly and not via Columbia/Baruch since these degrees would add little to your profile for PhD admissions. PhD admission Committees look for candidates that have demonstrated coursework that is theoretically rigorous in fields that relate to the research of their faculty and a clear indication of whether you've been sincere (yes, it's more important to be sincere rather than a genius who puts in little effort) with your studies over a long period of time. Both the Columbia MSOR and the Baruch MFE are professional degrees in the sense that they lay more emphasis on application of known models rather than the theoretical underpinnings and abstract properties of the model itself. Hence, they'd do little to augment your chances for PhD admission. However, if you must insist on getting a Masters before applying for a PhD, you're better of with Columbia any day since PhD admission Committees love to see degrees from known schools with stellar grades (I don't blame them since it's really hard for the Admissions Committee to admit, say, a Harvey Mudd graduate over a Columbia graduate even though the Harvey Mudd guy may have done better than the Columbia guy if admitted since the Columbia grad is "statistically expected" to be a more competent candidate. Impression (and not what you're actually capable of) is all that matters!)
Trading : If you do end up deciding to pursue a career in Trading, your Career performance is almost completely dependent on your Trading performance rather than the repute of your MFE awarding university or whether or not you have a PhD. Nobody really cares what your educational qualification is if you can't make a ton of money for them!
Quants : Yes, there are a quite a few Quant positions that are almost exclusively contested for by PhDs. Most of these positions can be found at hedge funds. I, however, know that this trend is changing and MFE as a degree is slowly getting accepted as a viable replacement since it prepares the candidate almost as well as a PhD does with regards to the requirements of the job and costs much less to hire. However, there are a few Quant positions that require you to have done research in, say, Artificial Intelligence, Bayesian Stats etc. specifically since the role requires you to be setting up models based on these concepts. Pure Math quants (i.e. quants who setup the math of new models and leave it to programmers to implement it) are also exclusively PhDs but these positions are literally in single digits and vacancies open up sporadically. If you aspire to be anyone of these guys, directly go for a PhD!
Otherwise an MFE/MSOR is sufficient. Also, choosing between them is a tricky call personally. If your basis of judgement is only pay, Baruch probably edges Columbia regardless of the degree personally. However, if you aspire to join a top 3-4 BB in a "prestigious hard to get" Quant role (and other prestigious hard to get roles) and are willing to go the extra mile and take on risk, Columbia is significantly better than Baruch. Also, since your an IIT alumnus, you can afford to take this risk since being one already has augmented your chances of getting into a top 3-4 BB so going the extra mile makes sense. I'm not ignoring the fact that Baruch MFE alumni have ended up at top 3-4 BB but they're less in number (even if a couple of BBs have started coming on Baruch's campus to recruit and in that sense have made it a "target" school; yet they recruit 1, maybe 2 candidates each out of the entire graduating cohort from what I can make out) than Columbia for sure which definitely makes a difference.
Hope this helps!
Good Luck.