New battery not recognized

I got hold of an old MacBook Pro (13" Retna Display Late 2013 - A1502) with a dead battery - the battery icon had a red "X" overlaid. I purchased an iFixit replacement battery - and the system no longer sees a battery at all. Everything else works (which means I reassembled the laptop properly).

I installed CoconutBattery 4, and it reports No Battery Found.

The Magsafe charger shows a green LED initially, which quickly changes to yellow.

But -

If I boot up on the magsafe charger, then unplug it, I get about 2 seconds ot lag time before the screen goes dark. If I leave it unplugged for a short bit (two minutes), then reconnect the Magsafe and press the power button to wake it, it picks up my logged-in session uninterrupted, which suggests that there's some power being provided from somewhere - else I'd be starting from a cold boot. (After 5 minutes. it's gone to screen saver witl password required - but my login session is still intact.)

I'm thinking (hoping) this is a matter of not recognizing rather than a hardware issue with the battery itself. Any suggestions would be appreciated.

Don't know how helpful this is going to be ... but the response is appreciated.