Odd power/battery startup problem on MacBook Pro

Question

I have a MacBook Pro 15-inch, Mid 2012 running OSX 10.9.5 with all updates.  Approximately one year ago the battery died and I'd been operating off the adapter ever since.  This past Thursday I finally had a Mac hardware guy from the local university install a new battery; something he does regularly for the university's MacBook Pros.  Weirdly, the old battery had no bulges or other visible problems.  He did an SMC reset, started up, and everything worked fine.  I later hooked up a brand new Acer external monitor mirroring it to the laptop, and charged the battery to 100% with the Adapter.  That night I disconnected the adapter and all peripherals without shutting down to have the battery drain somewhat overnight.  Friday morning the laptop started up with a partially depleted battery.  So, I hooked up the adapter and peripherals.  All was well. I shut down Friday night and restarted yesterday, Saturday morning, but there was a little glitch with the mirroring settings.  I solved that problem by deleting the Library/Preferences/com.apple.windowserver.plist file and restarting.  However, in trying several scenarios to make sure the display mirroring settings were sticking, I suddenly noticed the laptop would now not start up on battery power despite the battery being charged.  It would only start up with the adapter attached even when all peripherals were disconnected.
Today I have tried NVRAM and SMC numerous times with and without the adapter attached.  When the adapter is not attached the resets appear to be working (chime, grey screen, Apple logo) but then the laptop display dims for a second, then goes black.  The power light on the computer goes off.  So it appears the computer suddenly shuts down without completely starting up.  When I then attach the adapter, the computer starts up (and shuts down and starts up the exact number of times I tried either NVRAM or SMC or simply push the Power Button), finally starting up completely.
The standard Hardware Test (with no peripherals but the adapter attached) and the extended Hardware Test show no problems.
System Report contains the following info:
Battery Information:
  Model Information:
  Serial Number:    D861015004NDGDL3T
  Manufacturer:    SMP
  Device Name:    bq20z451
  Pack Lot Code:    0
  PCB Lot Code:    0
  Firmware Version:    406
  Hardware Revision:    2
  Cell Revision:    158
  Charge Information:
  Charge Remaining (mAh):    6704
  Fully Charged:    Yes
  Charging:    No
  Full Charge Capacity (mAh):    6910
  Health Information:
  Cycle Count:    1
  Condition:    Normal
  Battery Installed:    Yes
  Amperage (mA):    0
  Voltage (mV):    12383
System Power Settings:
  AC Power:
  System Sleep Timer (Minutes):    10
  Disk Sleep Timer (Minutes):    0
  Display Sleep Timer (Minutes):    10
  Wake on AC Change:    No
  Wake on Clamshell Open:    Yes
  Wake on LAN:    No
  AutoPowerOff Delay:    14400
  AutoPowerOff Enabled:    1
  Current Power Source:    Yes
  Display Sleep Uses Dim:    Yes
  GPUSwitch:    2
  PrioritizeNetworkReachabilityOverSleep:    0
  Standby Delay:    4200
  Standby Enabled:    1
  Battery Power:
  System Sleep Timer (Minutes):    10
  Disk Sleep Timer (Minutes):    0
  Display Sleep Timer (Minutes):    2
  Wake on AC Change:    No
  Wake on Clamshell Open:    Yes
  AutoPowerOff Delay:    14400
  AutoPowerOff Enabled:    1
  Display Sleep Uses Dim:    Yes
  GPUSwitch:    2
  Reduce Brightness:    No
  Standby Delay:    4200
  Standby Enabled:    1
Hardware Configuration:
  UPS Installed:    No
AC Charger Information:
  Connected:    Yes
  ID:    0x0100
  Wattage (W):    85
  Revision:    0x0000
  Family:    0x0085
  Serial Number:    0x004f99a7
  Charging:    No
Coconut Battery shows as follows:
Am I missing something?  Is there something else I can try so the laptop will operate on battery power only?  Thanks..

Answer 1

Step one, SMC reset:
http://support.apple.com/kb/ht3964
Step two, Apple Hardware Test:
http://support.apple.com/kb/ht1509
Ciao.