Thank you, that was a helpful suggestion. I made the log file and this is the relevant portion of the log:
File::transform:
Doing stat for mwstore://local-backend/local-thumb/9/93/How_Did_you_Hear_About_2014.png/120px-How_Did_you_Hear_About_2014.png
FileBackendStore::getFileStat: File mwstore://local-backend/local-thumb/9/93/How_Did_you_Hear_About_2014.png/120px-How_Did_you_Hear_About_2014.png does not exist.
BitmapHandler::doTransform: creating 120x68 thumbnail at /tmp/transform_79944211ec2f-1.png using scaler im
BitmapHandler::doTransform: called wfMkdirParents(/tmp)
BitmapHandler::transformImageMagick: running ImageMagick: '/usr/bin/convert' '-quality' '95' '-background' 'white' '/var/www/html/wikiname/images/9/93/How_Did_you_Hear_About_2014.png' '-thumbnail' '120x68!' '-set' 'comment' 'File source: http://intentionallyremoved/wikiname//index.php/File:How_Did_you_Hear_About_2014.png' '-depth' '8' '-rotate' '-0' '/tmp/transform_79944211ec2f-1.png'
wfShellExec: /bin/bash '/var/www/html/wikiname/includes/limit.sh' 'OMP_NUM_THREADS='\''1'\'' '\''/usr/bin/convert'\'' '\''-quality'\'' '\''95'\'' '\''-background'\'' '\''white'\'' '\''/var/www/html/wikiname/images/9/93/How_Did_you_Hear_About_2014.png'\'' '\''-thumbnail'\'' '\''120x68!'\'' '\''-set'\'' '\''comment'\'' '\''File source: http:// intentionallyremoved /wikiname//index.php/File:How_Did_you_Hear_About_2014.png'\'' '\''-depth'\'' '\''8'\'' '\''-rotate'\'' '\''-0'\'' '\''/tmp/transform_79944211ec2f-1.png'\''' 'MW_INCLUDE_STDERR=1;MW_CPU_LIMIT=180; MW_CGROUP='\'''\''; MW_MEM_LIMIT=307200; MW_FILE_SIZE_LIMIT=102400; MW_WALL_CLOCK_LIMIT=180; MW_USE_LOG_PIPE=yes'[thumbnail] thumbnail failed on star: error 2 "" from "'/usr/bin/convert' '-quality' '95' '-background' 'white' '/var/www/html/wikiname/images/9/93/How_Did_you_Hear_About_2014.png' '-thumbnail' '120x68!' '-set' 'comment' 'File source: http:// intentionallyremoved /wikiname//index.php/File:How_Did_you_Hear_About_2014.png' '-depth' '8' '-rotate' '-0' '/tmp/transform_79944211ec2f-1.png'"
[thumbnail] Removing bad 0-byte thumbnail "/tmp/transform_79944211ec2f-1.png". unlink() succeeded
User::getBlockedStatus: checking...
ForeignAPIRepo: HTTP GET: http://commons.wikimedia.org/w/api.php?aisha1base36=2d68fauliovh2w1k57br39hr2tsvmj7&aiprop=timestamp%7Cuser%7Ccomment%7Curl%7Csize%7Csha1%7Cmetadata%7Cmime%7Cmediatype%7Cextmetadata&list=allimages&format=json&action=query&redirects=true&uselang=en
LoadBalancer::reuseConnection: this connection was not opened as a foreign connection
[runJobs] Running 1 job(s) via '/wikiname//index.php?title=Special%3ARunJobs&tasks=jobs&maxjobs=1&sigexpiry=1416866196&signature=e8ce801451b14a3d680b953c8d298293fdffbfeb'
I then tried to run the command as you suggested:
me@place:/var/www/html/wikiname/images$ /usr/bin/convert -quality 95 /var/www/html/wikiname/images/9/93/How_Did_you_Hear_About_2014.png -thumbnail 120x68! /var/www/html/wikiname/images/9/93/bpthumb.png
convert: unable to open image `/var/www/html/wikiname/images/9/93/bpthumb.png': Permission denied @ error/blob.c/OpenBlob/2638.
convert: WriteBlob Failed `/var/www/html/wikiname/images/9/93/bpthumb.png' @ error/png.c/MagickPNGErrorHandler/1728.
me@place:/var/www/html/wikiname/images$
Helpful! So I then changed the permsions of ~/images and all subdirectories so that then MediaWiki user www-data has drwxr-xr-x I was then able to execute the code without an error and thumbnail file was correctly built as I can see when I downloaded it via ftp, but the MediaWiki still give me the same error code 2 and doesn't' produce thumbnails for the images (in say new files special page). I simplified the MediaWiki command to:
/usr/bin/convert -quality 95 -background white /var/www/html/wikiname/images/9/93/How_Did_you_Hear_About_2014.png -thumbnail 120x68! -depth 8 -rotate -0 /var/www/html/wikiname/images/9/93/transform_bp4.png
And I got:
convert: unable to open image `/var/www/html/wikiname/images/9/93/transform_bp4.png': Permission denied @ error/blob.c/OpenBlob/2638.
convert: WriteBlob Failed `/var/www/html/wikiname/images/9/93/transform_bp4.png' @ error/png.c/MagickPNGErrorHandler/1728.
So I suspect a permissions error, but I get the same error even when I set the ~/images/9 and ~/9/93 to drwxrwxr-x