There’s three kinds of strainers: Hawthorne, Julep and Fine mesh.
Straining Stirred Drinks
Julep strainers don’t really do anything except hold ice back. They’re great for drinks with no fine particles to filter out because they pour quickly, but you can just as effectively use a hawthorne strainer.
Straining Shaken Drinks
A hawthorne strainer has a spring that’ll catch more fine particles. They’re typically used for shaken drinks.
The only way to get citrus pulp and tiny ice shards out of a drink is to double strain. Use a hawthorne strainer on the tin and pour through a fine mesh strainer.