As it has been pointed out already, the more exotic metals and removal can and will be a problem in an emergency, however, there is a minor exception. Commercially pure grade 1 (CP1) titanium only has a top tensile strength of 35ksi and yield strength of 25ksi with CP2 not too much harder. It's the CP3 or higher impure or titanium aerospace alloys, the tungsten alloys, etc. that are really dangerous and really difficult to remove at 5x-plus tensile/yield strength by medical staff. For a comparison, annealed sterling silver is 35ksi tensile and 18ksi yield, and even the softest stainless steels are still much harder to cut and work at twice that strength.
Another thing to watch for with the cheap rings is an inner pipe cut, which also can make the ring much harder to remove under... circumstances. You want a fully internally rounded comfort fit interior, properly sized to a 1/4 ring size, where only the peak of the inner curve is your actual ring size with the remaining inner surface larger than your finger. A CP1/2 titanium ring still might not be able to be easily resized smaller, but this permits a little bit more weight variability and makes the ring considerably easier to remove and even gives a little room to mill the ring a bit larger. Unfortunately, the only way to guarantee getting the more expensive pure titanium is to deal with a reputable jeweler who actually makes the things. Fortunately, the trivial additional cost over the mass produced stuff (expect to pay around $100) gives you the opportunity to customize the design to an extent.
Keep in mind, even though aerospace titanium alloy is harder to mill and work, it is far cheaper to produce in bulk. Price reflects this, and the harder to remove alloy rings will be far cheaper than the pure titanium.
At the end of the day, though, remember... it's just a symbol, and still just stuff. Invest yourself in what it represents, not in the item itself.