Your second and third methods can never match, as none of your target strings begin with, or contain only IF-D. The second method uses ^, anchoring to the beginning of your target string, and the third method contains both ^ and $ mandating that the pattern match the entire target string, not just some portion of it. So those will always fail (it appears that you're just trying things at random to see if they work, and especially in the case of regular expressions, that's not a good way to accomplish the goal.)
The first example will match one time; the 2nd element, because the pattern IF-D matches at the end of the target string Link-IF-C<->IF-D. However, it's only going to work if your target string and your pattern are what you think they are. In the example code you showed us, the pattern string wasn't wrapped in quotes. It must be.
So, for example, this will do what you seem to want:
my $link = "IF-D";
my @array = qw(
Link-IF-A<->IF-B
Link-IF-C<->IF-D
Link-IF-E<->IF-F
Link-IF-G<->IF-H
Link-IF-I<->IF-J
);
my $found = grep /\Q$link/, @array;
print "$found\n"; # 1
The \Q isn't strictly necessary for the pattern you've demonstrated. That construct forces the contents of $link to be treated on its literal meaning, rather than possibly as metasymbols. Your example pattern doesn't contain any metasymbols, but if it accidentally did, the \Q would de-meta them.
If you think you've implemented something semantically equal to this example, and yet it's not working, then you've found exactly why people ask that those asking questions post a small self-contained snippet of code that demonstrates the behavior they're describing. If my example code doesn't clear up the problem, boil your code down to a single snippet that demonstrates the problem, and add it as an update to your question so that we can run it ourselves and see exactly what you're talking about.