Before I forget, I recommend sealing the wooden floor before applying the vinyl. Just a double layer of protection.
What Minwax Polyurethane do you have exactly? If it's oil-based I would proceed with extreme caution. With current VOC regulartions Minwax and other companies have had to greatly reduce the solvent concentration in the oil-based products. It's a long explanation, but this makes it very fussy to apply and actually increases how long the product offgasses. I know it seems backwards, but less solvent in these products means they actually release solvent longer. The reason being is that the coat goes on too thick and a thick skin is formed and the solvent can't escape properly. Not good for a snake cage.
You'll be able to tell if it's oil-based by what they recommend for thinning it and/or for cleanup. If they recommend mineral spirits, turpentine, or basically anything other than water then it's oil based.
Oil based polys offer an incredible amount of protection for the money but they offgass for years. No kidding. You're not even supposed to use them on the inside of kitchen cabinets that will store dishes. I just don't recommend them anymore for reptile cages. Also, they turn a yellowish color when they cure that looks terrible on light woods like your russian baltic birch. They are better suited for darker woods.
But back to your question, I think a water-based polyurethane would be best but I'd recommend covering the first few inches of the cage walls with vinyl flooring. This will be tough with your interior frame. A WB poly will dry clear and look great on the birch.
If you absolutely must go to a Home Depot like store then I think Varathanes WB poly is better than Minwax's Polycrylic.
But in general woodworking specialty stores will have better products than Home Depot or Lowes.
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Current snakes:
0.0.1 Gonyosoma oxycephala - Java locale (green)
2.2 Gonyosoma janseni - Seleyar locale (all black)