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Asset can also mean under influence of Russian people with Kremlin connections.
The autobiography of former KGB chief Oleg Gordievsky is interesting in this regard. A lot of his early career was just taking his assets out to dinner, get them drunk and elicit tiny nuggets of info each time. There seems to have been few major scoops, mostly small pieces of the puzzles here and there that ultimately proved useful.
In that light Johnson’s unsupervised cajoling with the Lebvedebs is extremely unfortunate. Enough to call him an asset IMO.
Also, to be classified as an asset you don't have to be a full on spy. It's enough if the KGB/FSB officer is able to reclaim the dinner/champagne/escort on their expenses. I doubt the Lebvedevs will claim for each bottle of champagne Johnson drank during the visit to the villa in Italy. But I don't think it's far fetched at all to assume they were rewarded in some way or another for having had the UK foreign minister come visit.
Depends on what people mean by asset.
Asset doesn't necesarily mean on FSB payroll and conspiring with Moscow. Asset can also mean under influence of Russian people with Kremlin connections.
If you use the latter definition, then I think its fair to say that Johnson is an asset of Russia. Whether he even realises this himself or not. Another way to describe him would be useful idiot instead of asset or spy.