Edited 2017 - changed recommended long life storage voltage and added comments on fast charging using some recent systems. RM.
What YOU do as regards several of these questions depends largely on what YOU are trying to achieve or test.
Discharge to cutoff is fully discharged (to whatever remaining % that voltage represents). That's the easy one :-)
Percent dropoff of current in tail sets final % of max possible charged reached. There was a superb table given here within last week or so. Can supply later if you don't find it.
Real Men™ plateau at 4.2V and tail down to 10% or even 5% of the constant current rate. This gets the battery full and knocks the stuffing out of it.
Others terminate the current tail at say 25% of cc value.
Optimum lifetime for ongoing usage is at about the end of the constant current phase. That makes it very easy to locate - charge at specified current until desired max voltage is reached, then charge at constant voltage as desired. Here "desired" is to stop immediately. This is the point at which batteries tend to give significantly longer whole of life mAh of storage without grossly reducing mAh capacity per cycle. This is liable to be the point where older "fast chargers" tell you they have finished. Actual % total claimed varies but probably 70% - 80% range.
Newer USB input fast chargers use the term differently. In the case of USB the maximum available charge current at 5V is 5A so that the battery MAY be able to be charged at ~= 6A for the CC part of the cycle using an efficient buck converter to drop voltage and raise current.
[For a buck converter: Vout x Iout = Vin x Iin x efficiency_of_conversion]
Some systems such as QuaqlComms Quick Charge system allow the use of higher charger voltages (9, 12, 20) with specifically designed equipment, so battery charging can be faster for a given voltage provided that the battery specification allows this.
Maximum charge rates for LiIon and LiPo batteries are usually C/1 = 1A per Ah of battery capacity.
At 5V, 5A a USB charger can charge a 6000 mAh 1 cell LiPO battery at max rate - so eg a 10,000 mAh single cell battery used in some larger tablets can not be charged at the allowed 10A ! rate.
For long life storage where actual stored capacity is unimportant, LiIon and LiPo cells should be stored at about 3.7V.
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Using cells without protection adds to the rich tapestry of life. As long as you don't mind the occasional scorch mark on the tapestry that's fine. Note that part of the protection is a one time high capacity fuse under the cap for when things get out of control. Undervoltage discharge destroys. Charging from below a certain voltage at full rate can get fun, I'm told. Charging at reduced rate can bring cell up, I'm told. Below another second level they say don't even think about it. I've had very poor success in trying to get LiIon to misbehave. I have a box of unprotected cells that are very uncooperative about venting with lame etc. Strange. Sony and Apple and even HP seem to be much better at it :-).
I think what they're trying to do here is 'trick' the phone's battery charging intelligences. Li-On batteries are very touchy and have somewhat complex charging strategies. It all boils down to determining something called State-Of-Charge (SOC). SOC is just a percentage in the end, but arriving at the SOC number relies on a large number of factors that are not always easy to read and sometimes must be indirectly inferred. For instance, let's assume that you have a cell phone with a Li-On battery that is 3.7V and 1000mAH. We'll start with it being fully charged, so we know SOC is 100%. As you use your device you're drawing current out of the battery and the battery's voltage will drop - eventually. By measuring the current and monitoring the voltage you can guess what the SOC is. One problem is that the voltage isn't terribly useful in determining SOC because it doesn't change very much until the battery is nearly empty - that is NOT something you want to do to a Li-On battery. So you're mainly relying on the current.
So your SOC is being estimated throughout the usage. It gets low - 50% maybe - so you plug it in to charge. While it's charging, it monitors the charge current and battery voltage to determine when SOC is 100% again. Only, due to errors in measurement it says that the charge is complete when SOC is actually only 95%. Now your phone thinks 95% is fully charged - and it remembers this for future reference because it doesn't want to over-charge the batteries (this is also very bad). So essentially it's trying to read when the battery is full by measuring what goes in and guessing where that puts the SOC based on past results.
The errors aren't large so during normal charge/discharge usage you won't notice a problem. But sometimes the errors can stack up and your phone thinks its fully charged when it has little or no charge - it goes straight from full to empty and due to the incorrect SOC calculation the phone won't try to charge the battery more because it doesn't want to damage it.
In these cases you have to reset the SOC state. I have a Droid Incredible 2 and I've done it by removing the battery and holding the power button for 30 seconds, then putting the battery in and charging the phone while its off. This always fixes the issue where the battery thinks its full but drops down to something like 10% very quickly and the issue where it thinks its at 10% but has much more charge left.
The strategy outlined in your post is obviously trying to recalibrate the SOC or trick the algorithm somehow. Having never developed a charger that relied on SOC I can't say whether it will work but it seems like a lot of effort for a questionable amount of benefit. If your battery is acting really funny try what I suggested first.
Best Answer
Your cell is not charging more quickly than normal. Rather, the other battery in your graph is charging much more slowly than normal because it has high internal resistance so quickly reaches the CV = Constant Voltage phase, which means that the charge time will increase because more time is spent in the CV phase (at lower charge current).
Below is typical CC,CV charge for a similar capacity cell. Notice that the charge completes in 73 mins - close to the 80 mins of your 430mAh cell. Notice also that it stays in CC = Constant Current mode for 57 of 73 mins, about 79% of the charge, which is typical for a healthy cell.
Contrast that to the graph you supplied below, where the 1C charge quickly enters CV in about 12 mins of a 141 min charge, i.e. at about the 9% time mark. This is typical of a very unhealth cell with very high internal resistance.
The first graph is from the site lygte-info.dk, which has a large number of reviews of batteries and chargers. Perusing some of those should give you better intuition on typical (dis)charge curves.