What I would do is cut the downspout above the barrel. Use the existing fittings as shown in picture to allow the flow to go into a hole cut in the top of the barrel.
Are you going to use a pump to pump water out of the barrel or add a spigot at the bottom to use a garden hose?
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I think Anne is on the right track. Eavestroughs and downspouts are notoriously non-standard. I often modify square / rectangular tube with an open notch at the end to interface with ... anything. Or I will add temporary open eavestroughs that I have scrounged.
These are perfectly functional and free solutions, but not exactly neat-and-tidy from an aesthetics point of view. There's a fair bit of Heath Robinson / Rube Goldberg in my water catchment systems.
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