Backspace-and-retype a list of words

Perl, 43 bytes

42 bytes of code + -n flags.

chop$@,say$@while!s/^$@//;s/./say$@.=$&/ge

To run it:

perl -nE 'chop$@,say$@while!s/^$@//;s/./say$@.=$&/ge' <<< "abc
abd
aefg
h"

Java 8, 144 bytes

This one is similar to the reference implementation but combines the two while loops. It's a lambda expression accepting a String[] parameter.

a->{String c="";int l=0,i;for(String w:a)while((i=w.indexOf(c))!=0||!c.equals(w))System.out.println(c=i!=0?c.substring(0,--l):c+w.charAt(l++));}

Ungolfed

a -> {
    String c = "";
    int l = 0, i;
    for (String w : a)
        while ((i = w.indexOf(c)) != 0 || !c.equals(w))
            System.out.println(c = i != 0 ? c.substring(0, --l) : c + w.charAt(l++));
}

Acknowledgments

• -38 bytes thanks to CAD97's lambda suggestion

Pyth, 25 23 bytes

V+QkWxNk
=Pk)
M>._Nl~kN

Try it online.