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Were your contacts Google Contacts? if so, they may simply be stored on the Google servers, and be retrievable from there.
If not, then the next question is, does your Motorola cell phone still work well enough, that it will power up if connected to a computer?
If not - try it and see, simply - then you cannot get contacts or other info off it that way.
...unless it has them stored on a memory card plugged into the cell phone (- if it is an old cell phone, I doubt it has a memory card storage ability.)
Let me know the model # ***** at least the model name of the old Motorola cell phone. It might suggest something that can be done if all the above cannot be done.
Is there anything on the outer case of the phone?
The battery is non-specific, I can't tell anything from it. It's just a battery.