I have found lots of contradictory info on the web about the
difference between LR44 and SR44 button cells. Or just LR and SR
cells.

(1) Some say that the LR44 is ALKALINE, others say it is SILVER
OXIDE. Some same it is both!

It is alkaline. A lot of people are confused nowadays.

(2) Another says LR44 and SR44 are designations used by DIFFERENT
MANUFACTURERS. For example: http://www.tic-toc.co.uk/chart.htm

No, they are the same mfr., alkaline and silver oxide respectively. At
least that was the case when I was researching camera batteries about 5
years ago.

(3) Some say that a silver oxide cell has greater CAPACITY than the
equivalent alkaline. http://www.sr44.com/

Depends on the load current, probably.

(4) Others say that the silver oxide SELF-DISCHARGES quickly other
say slowly. http://members.iinet.net.au/~fotoplot/accbatcc.htm

Slowly.

(5) Some refer to RATE OF DRAIN and say the silver oxide can take a
heavy load. Others say the alkaline can deliver more.

Hard to say unless quantified. Also depends on whether you are willing for
the voltage to drop as the battery gets used up. Alkaline drops a lot more
than silver oxide.

(6) One says that some SR44s are EQUIVALENT to type 357 but that some
different SR44s are equivalent to type 303. Weird.
See http://snipurl.com/lmcf

When I looked into it, 357 was an SR44 with lower internal resistance than
an MS76, which is also supposedly equivalent to SR44. Olympus cameras tend
to prefer 357 to MS76. (OM-2S and OM-4T SLRs, that is.)

(7) Most seem to say that the silver oxides deliver a constant
voltage until they pack up whereas the alkalines deliver a declining
voltage.

Right.