Sorry for late , i know the frustration of a project waiting...lets get into it
There are 2 types of lithium battery charging module available in market as you shown both uses tp4056 cc cv but one (the larger one consisting 2 chips) has 1 extra tiny ic for auto cut at low voltage generally at 3.3 v to 3.5 v it can be adjusted if you remove 1 of the smd resistors and replace that with potentiometer..any way were getting out of topic...if you have any questions regarding that fell free to ask...
Only the first image supports auto cut
Heres how to connet it...you can see the output has 4 wholes 1 pair + - and 1 pair B+B-
Connect the B terminals to battery as mentioned and the load to adjucent +- terminals the and you may keep the circuit plug in through usb or you may directly attach wires through top wholes..
I wish your question has been answered, let me know if i missed something