Where I am now (17Jan13). The section "POSTSCRIPT PRINTING RENDERER" is no longer to be found in "README - OpenPrinting CUPS Filters v1.0.11 - 2012-03-29". So I'm stuck there.
I installed the debs from Brother and modified the printer version in CUPS using Brother supplied drivers, but, alas, no improvement to report: # dpkg -l | grep "Brother.*5370" ii cupswrapperhl5370dw 2.0.4-1 i386 Brother HL5370DW CUPS wrapper driver ii hl5370dwlpr 2.0.3-1 i386 Brother HL-5370DW LPR driver Printing the pdf files generated by Iceweasel does, indeed, appear to be a separate issue. Even pdf's generated by pdflatex do not print properly. Something of a mess. One positive note: I was able to print the problem web pages from Iceweasel on the HL-5370DW using the pxlmono PPD, a stop gap that I am using on an older Brother HL-5??? (don't remember exact model) as well. Thanks for the responses. I appreciate it when people take their valuable time to read and respond. Mark