Water ingestion can lead to a flame-out. That's the biggest hazard. If it's cold water (eg, supercooled), inlet icing and even stator, fan, or compressor icing is a hazard. The recent spat of Beechjet dual engine flameouts is a good example of what can happen.
The turbofan is far less susceptible to the hazards of large amounts of water passing through the engine, as most of the water goes around the core.
If you're flying in an area of extreme precipitation, often the water is not the biggest hazard with which you must contend...and you probably ought not be there.
Ignitors on in heavy rainfall or precip is always a wise choice. While many aircraft have autoignition, many don't. If it gets heavy, going to manual ignition may be a wise choice.
Keeping engine speed up in heavy precip is also a plus for your engine. It's physically easier on the engine, reduces the AoA for each blade (and any attendant physical damage or icing) creates a larger and more rapid temp rise in the engine (compressor and turbine), and in general, makes it harder to put the fire out.
So far as static or rolling, neither one is particularly relevant so far as water ingestion, except that some engines at high power will "suck" water from the surface. Water ingestion as moved from the nosewheel is a big issue and could cause a flameout. In either case, static or rolling, the aircraft will have to transit the water at the same speed at some point in the takeoff, and will be exposed to the same levels of spray. If one area of standing water exists differently from the rest of the runway, then passing through this area slower is a plus, but not at the expense of runway length.
Standing vs. rolling is an issue of takeoff distance, in which case the aircraft manufacturer proceedures and company proceedures should be followed closely. A wet runway reduces one's ability to stop on an aborted takeoff, and may lead to hydroplaning issues on the go or the stop. Consequently, the amount of pavement ahead is really effectively diminished with respect to stopping the aircraft. If the field is critical, which is not only a function of runway condition in this case but weight, altitude, and temperature, holding brakes and running power to a predetermined value before release is the only way to gaurantee performance numbers.
You cited the case of two aircraft of the same type departing one after the other...but were they at the same weight? Probably couldn't tell, but that does make a difference.