MacBook Pro only charges with smaller MagSafe 60W

Try to do a SMC and PRAM reset.

SMC, amongst others, controls power management of your Mac. From https://support.apple.com/en-us/HT201295:

These symptoms might mean that an SMC reset is necessary:

• [...]
• The battery doesn't charge properly.

You may or may not get any useful information from pmset, but it takes virtually no effort, so why not?

• What is pmset? A command line utility for manipulating and viewing power management setttings under macos. Ref Wikipedia article

• You can access pmset from a terminal window. You can access the system manual for pmset by entering man pmset at the command line (type q to close the manual).

• The -g options for pmset will display potentially useful information, and do not require root/su privileges. You will be required to use root privileges with pmset to make changes to system settings, but that may not be necessary at this time.

• You may wish to try the following when your chargers are connected, and note the differences. The output below is from my MBP:

pmset -g ac
 Wattage = 86W
 Current = 4300mA
 Voltage = 20000mV
 AdapterID = 5718
 Manufacturer = Apple Inc.
 Serial String = C047*************       # * replaces a portion of this string
 Adapter Name = 87W USB-C Power Adapter
 Hardware Version = 1.0
 Firmware Version = 01070069
$

• You might also try monitoring the log for a bit to see if that turns up anything relevant:

$ pmset -g rawlog
pmset is in RAW logging mode now. Hit ctrl-c to exit.
 * Battery matched at registry = 7939
05/11/19 23:01:34
 AC; Not Charging; 99%; Cap=6154: FCC=6181; Design=6669; Time=0:00; 0mA; Cycles=286/1000; Location=0; 
 Polled boot=05/11/19 14:56:06; Full=05/11/19 22:54:49; User visible=05/11/19 23:00:49
05/11/19 23:03:49
 AC; Not Charging; 99%; Cap=6155: FCC=6181; Design=6669; Time=0:00; 0mA; Cycles=286/1000; Location=0; 
 Polled boot=05/11/19 14:56:06; Full=05/11/19 22:54:49; User visible=05/11/19 23:03:49

Note that the charger was connected to my MBP during this logging activity, but the battery showed 100% charged in the menu bar at the top of my monitor.

Finally, if you see anything in the pmset output that seems suspicious, please consider editing your question to include that. No promises any of this will help of course, but we'll certainly give it a try.