Things to consider:

How many miles and years are on this alternator? If over
about 85k miles, consider replacing the alternator brush
assembly. Plenty on the net to document that worn brushes
tend to be the most common cause of alternator failure and
are the essence of an alternator rebuild and will cause the
warning light to come on and go off intermittently.

An OEM brush assembly may be had at www.slhonda.com at a
very fair price.

It could be a loose belt, as well.

If the alternator is old, I'd replace the bearings, brush
assembly, and belt, tighten the belt per guidance on the net
and see what happens. That screech at cold start is likely
the belt and is exactly what I had not long ago when I
replaced the belt by myself for the first time. Folks here
said to tighten it. That was the problem.

I trust you have thoroughly cleaned the battery terminals
and tightened its connections.

--------------------------

You should have changed the BRUSHES while it was out. That's the only
part that wears out (unless you mess up the diodes while boosting). Too
bad Autozone didn't suggest a brushes kit instead . . .

'Curly'

In the process of fooling with it you dislodged the brushes and it
started working again.  When a brush is at the end of the service life
it has trouble maintaining the proper contact.  Often you can tap the
alternator when it is signalling no charge and it will start working
again.  It will, however, go permanenetly bad within a relatively
short period of time and you will need to replace the alternator or
re-brush.  

I limped along with a bad alternator on a 90 for about four months by
giving it a sharp rap each time it was signalling no charge.

Frank

It's the brushes.  

I just went through the same thing on my '87 Si.  Same symptoms.  Pull
the alternator again ...it'll be easier this time... and replace the
whole brushes assembly.  You can buy a pair of new carbon brushes for
less than $5, but you have to tear the brushes assy apart and resolder
the brush leads and get the springs to fit back.... SCREW ALL THAT!!.
Just buy the complete brush assembly for about $30 from your dealer
and it's a screw/nut driver replacement job.  Shouldn't take more than
a couple-three hours.

Tip: When installing the new brush assy over the shafts, insert a
thin metal pin or rod between the shaft and the brushes to hold the
brushes up off the shaft.  When the assy is fully seated, pull the pin
to let the brushes drop.  I used my smallest allen wrench.  

nb

Its the brushes located on the end of the rotor.  I went to the
wrecking yard and recovered a  half worn set by taking along a few
small wrenches and removing the stamped metal cover on the rear of the
alt.---while still on the car.  Mine lasted for 192,000 miles (1991
Accord EX) and the set I recovered was from a 1996 accord.  Same
thing---half worn---good enough for me.  They wanted hundreds at the
parts places for a rebuilt alternator. Works fine now, but I did buy a
new alt. and power steering belt and replace them while I was at it.