Easiest solution is to use “mmv”
You can write:
mmv "long_name*.txt" "short_#1.txt"
Where the “#1” is replaced by whatever is matched by the first wildcard.
Similarly #2 is replaced by the second, etc.
So you do something like
mmv "index*_type*.txt" "t#2_i#1.txt"
To rename index1_type9.txt to t9_i1.txt
mmv is not standard in many Linux distributions but is easily found on the net.