The cell probably has a 4.3v or a 4.35v charge termination voltage.
Samsung didn't invent a new chemistry, but there is ongoing refinement of electrode composition and electrolyte additives in pursuit of some combination of higher energy capacity, higher power delivery and lifetime. At this point, 4.35v cells can deliver higher energy capacity while still having similar lifetimes to those considered acceptable with older 4.2v-termination cells.
There is no harm charging it to 4.2v, other than leaving ~10-15% of current capacity unused (and slightly more of the power capacity) until you can give it a proper charge. Similarly, there isn't any harm in just terminating charging immediately at 4.2v or whatever the proper termination voltage is, the only downside is, again, leaving some capacity fallow.
As noted, most TP4056 boards these days seem to have knockoff chips on them. Fortunately, the knockoffs I've heard of people testing seem to perform similarly enough. Of course, who knows how consistent they are.
I wish that modules for higher termination voltages, or even better, adjustable termination voltages, were easier to come by, even at 2-4x the price of the TP4056 boards.
Oh, following up on Rick Law's suggestion of checking the actual charging current if you replace the resistor (or even if you don't), all these battery charging chips are pretty sensitive to resistance between charger and battery, so sticking a multimeter between the charger and the cell can mess with the measurement. On the other hand, the TP4056 and clones are all linear charge ICs, and don't draw all that much current on their own, so you can get a reasonable estimate of current going into the battery by measuring the supply current.
Finally, while yes, the leads protection circuits, etc all result in some voltage drop, a lot of it is related to the amount of current. So, while the charger might start ramping down from 0.5A before the battery is at 4.2v, as the current drops, the difference between the cell voltage and the voltage the charger is reacting to will drop, and by the time it is near termination current, the resistive losses are going to be pretty small.