Still here.
You've basically nailed it. The transient response of a SDC is much faster than an LDC. If you're using overheads for 'just' cymbals and the attack of the toms, then a pair of SDCs are ideal. If you're after an overall kit sound, then an LDC is more appropriate.
The difference is quite small and I've used LDCs and SDCs in both applications. In a live setting, a set of SDCs is usually more appropriate to avoid frequency 'clustering' (usually around 250Hz - where most instruments have some kind of output) and avoiding stage rumble. A lot of decent LDCs have a low-frequency cut (shelf) filter to avoid stage rumble. High-quality LDCs also frequently have multiple polar patterns - which is a more obvious difference. AKG C414s are what I would think of as a 'standard' studio mic - with all of the above options.
In a studio, you can EQ the extraneous low frequencies out more easily and you'll get a more natural sound with an LDC. Having used a set of AKG C214s in the past, they would be my choice within the models you've mentioned - even though I own a C1000S (for its ability to be charged with a battery).
As with everything recording, placement is much more important than the mic you are using.