No, your song represents an emotion; longing. Many songs do that.
A story, at its minimum, is a sequence of causes and effects, typically leading the listener to wonder what happens -- and then get the answer.
It does not require an antagonist, really, or human conflict.
In a little while from now
If I'm not feeling any less sour
I promise myself to treat myself
And visit a nearby tower
And climbing to the top
Will throw myself off
Gilbert O'Sullivan: You have a story in the first lines: The singer has made a conditional promise: If they don't feel better soon, he will "treat himself".
The listener is engaged and wondering what will happen. This is a story.
… "Let us be lovers, we'll marry our fortunes together
I've got some real estate here in my bag"
So we bought a pack of cigarettes and Mrs. Wagner pies
And walked off to look for America
Paul Simon: Bingo, a story in the first line: A proposition is made, "Let us be lovers and marry our fortunes together".
Will it happen? Yes, we get the answer in a few lines, but then WHAT will happen? The song is the story of a desperate search for a place to belong. (IMO).
Many songs are just about a momentary feeling, love, or sorrow, getting fed up and breaking up, struggle, etc. You could call these "declarations". I feel strong. I feel victorious. I feel angry. I feel healed. I feel saved.
Story songs are often also about a theme feeling, but told as a sequence over time. "A" happened, so "B" happened, so "C" happened.
In the song "Alone Again", the song theme is loneliness, but told in a sequence: We begin at the end: He is stood up at the altar and this is the last straw, he is suicidal. Then we flash back, to all the times he has lost people and felt alone, time and again.
… I learned the truth at seventeen
That love was meant for beauty queens
And high school girls with clear skinned smiles
Who married young and then retired
Janis Ian. Again: Story from the first line: **What is the truth?!?!"
None of these songs require interaction with others; Paul Simon's has a lover but they are not in conflict, they are fellow searchers taking comfort in each other.
All the songs have an emotional theme, we recognize it whether we can name it or not.
The difference in story songs is they relate events that brought on this emotion. Why is Gilbert O'Sullivan feeling suicidal?
Why is Paul Simon feeling lost and melancholy?
Why is Janis Ian feeling unloved, ugly, left out?
For your "Old West" song to be a story, the emotional THEME would be you want the Old West back, the STORY would be WHY you want the Old West back.
What happened to you?
Or what is happening to you?
You have 2 or 3 verses and a chorus, what are the 2 or 3 life problems that would be solved if you were back in the Old West?
What was allowed or prevailed then that you wish was allowed or prevailed now?
Duels? A simpler life? Riding the open range and living under the stars? Fresh air? True love?
Pick your theme, the feeling you want to convey. Then your story (in poetic verses) is the 2 or 3 most important incidents that have made you feel that way.