Millssa,
If the terrazzo is cement terrazzo (which is what I would expect in 1909 building) then the historically appropriate treatment would be to grind the terrazzo with a series of progressively finer grits to polish the surface. The polished surface is quite durable to foot traffic, which is why the material was used so much in public spaces, and is relatively dense and resistant to staining. Of course, the challenge is finding a specialist contractor who has the appropriate equipment and can perform the work; they are becoming comparatively rare. Contractors that install the resin-based terrazzo systems in common use today may have experience with cement terrazzo, but you should confirm and ask for references.
Once the terrazzo is polished, an acrylic finish designed for the purpose can be applied to protect the polish, but resin finishes would not have been available at the time the floor was installed, and they do require periodic renewal. They have the benefit of being able to confer slip resistance when selected for that property. The method your maintenance staff has been using is typical for terrazzo floors in public spaces. Using mats or some other type of physical surface barrier for the traffic areas might be a lower-maintenance approach to maintaining the condition of the floor once it is re-ground (a linoleum carpet would be period-appropriate). Such a covering also provides slip resistance for visitors in walking areas. Regular maintenance is required in the form of dust mopping and mopping; if a physical barrier is used, it should be lifted and cleaned under to prevent trapped grit from damaging the polish.
With regards to your marble floors, there are more questions there because the installation details are important to selecting an appropriate approach. My recommendation would be to reach out to an architectural conservator that has specific experience with interior floors.
Kind regards,
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Lorraine Schnabel
lorraine@schnabelconservation.com------------------------------