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It sounds like you're describing two different fits honestly. To me a modern relaxed pair of trousers means borderline baggy - and most manufacturer's think the same thing based on the silhouettes being sold under 'relaxed.' Whereas a carrot-ish fit just sounds like what they describe as 'tapered'.
You may want to look up athletic fit, which is just a wider top block with a taper below the knee. Or a straight slim which is just a narrower straight fit that follows the shape of a leg more - rather than a traditional straight fit which is straight from top to bottom.
Don't buy Levi's though. High street mass produced trash jeans, nowadays.
Where should I be looking for some jeans that have a more relaxed cut as is becoming de rigueur?
Possibly a sort of carrot fit to just above the top of the shoe.
Do Levis do a cut that fits the bill?