I guess a 'girder' in an overhead travelling crane is a horizontal beam, which supports the path for travelling crane i.e. a crane when moving travells through the girder?
www.jherbertcorp.com/crane-bridge.htm has a good set of diagrams that should answer your question. The terminology "bridge girder" and "runway girder" are consistent with the terminology and definitions used in the 1967 edition of the Whiting Crane Handbook, a widely recognized source of crane design and application information, no longer published. There is also a definition of "auxiliary girder" such as may be used to support footwalks and the operator's cab on a large crane bridge.
Hard to tell whether you are referring to the bridge girder or the runway girder. On larger bridge cranes they mostly seem to run the trolley rollers on rails mounted on the upper side of the top flange of the girder. But note the first illustration of a small crane in my cited link shows a simple trolley running on the inner side of the bridge flange. ("through the girder"?)
BTW, I'm not a crane expert; just a mechanical engineer who found several rare copies of the Whiting Crane Handbooks to use as references for crane models for an HO model train layout.
There are many different varieties of
cranes on the market today, both in terms of size and type. Each type
and size serves a different function. In essence, a crane is a
machine, which lifts materials up and down as well as horizontally.
They are most often used in the construction, manufacturing and
transport industries.
Cranes are typically used to move and
assemble heavy equipment. Crane sizes run the gamut from the enormous
ones all the way to much smaller ones which tend to be used in
factories and workshops. There are also really tall ones typically
used in the construction of tall buildings.