I have bought a Li-po charger the other day and I looked up the IC and it stated Li-ion. Can you charge a Li-po with a Li-ion IC? And the other way around? The IC I am talking about is the TP4056. I thought Li-po is quite difficult to charge because if your voltage is to low it does not charge anymore and if it is to high you can blow it up.
For charging purposes, the two types (Lithium Ion and Lithium polymer) can be considered equivalent.
Both types have the same issues you noted; If the initial voltage is too low, it will very likely never recharge, and the terminal voltage must not exceed 4.2V or pyrotechnically interesting things can happen.
I agree with the answer given by Peter Smith in that the Lion and Lipoly can be considered the same for charging purposes. They certainly use the same algorithm.
However, I own 3 different battery chargers that charge all types of rechargable batteries. They all make a small distinction between the Lion and Lipoly in terms of their voltage. For LiOn, they claim a nominal voltage of 3.6v and a termination of charge at 4.1v. For LiPoly, the nominal voltage is 3.7 and terminal charge voltage of 4.2v.
The datasheet of your battery will tell you which voltage you should charge to, 4.1 or 4.2. But it seems like all lithium batteries in the last 5 years have been 4.2v for max voltage and LiOn and LiPoly are used interchangeably.