It sounds like you're confusing a couple of things here.
Cut in refers to the point at which a generator begins to actually produce power.
Remembering Ohm's law is critical to understanding this.
The pertinent formula for this particular concept is P=IxE
Power (Watts) = I (Current) x E (Voltage, or potential)
A spinning generator with nothing connected to it produces a potential proportional to the speed at which it is rotating, but does not produce any power, because there is no current flow since the circuit is not complete.
In order for current to flow, you must have a complete path, and the potential to push this current. Voltage is pressure, just like in an air tank. Current is the number of electrons passing through a conductor.
Combined using Ohm's law, this is the Watt; the measurement of how much work is being done. In the case of a light bulb, it is how much energy it takes to get the filament hot enough to get the rated amount of light out of the bulb.
Exerting energy makes you tired because it's more work than doing nothing right?
Therefore, with no current flow, no work is being done, and no energy is used. Until you reach cut in, no current is flowing. The shaft of the generator is easy to turn.
At cut-in, the generator's shaft will begin to provide significant resistance to being turned any faster.
With that said:
The level of charge on the battery determines the cut in voltage. Any time the generator would have a higher open terminal voltage (OTV) than the battery, the generator will see a load, and thus suddenly become more difficult to turn at that RPM.
If a low battery is connected to a generator, the generator will reach cut-in at a lower RPM than it would if the battery were full.
This happens with a purely resistive load as well (such as a light bulb), but gets overlooked because the generator will always have a higher potential than the light bulb when it is turning. This is why no matter how fast you try to spin the generator with the light attached, it is always more difficult than if there is nothing connected to it. A light bulb always tends toward a potential of zero between the terminals, since it has no way of storing energy electrically.
Explaining grid tie gets hairy; as it would have several other mechanisms in place that would obfuscate cut in, but the same basic principles still apply.
In a nutshell, if the generator is not spinning fast enough to make a high enough voltage, the grid will remain above it, and the generator will never see a load. Once the generator is spinning fast enough to produce a higher potential than the grid, current would flow, and the generator is said to have reached cut in.
Fair warning, however: There is much more to consider than this when dealing with the grid, however, such as frequency, phase, syncronization, and so on, so please don't try any experiments like this with the live grid, lest you don't value safety and the like. AC is a vastly different animal than DC, particularly when combining power sources.
Steve