Hallo, Thea. White Star's wine list at that time included three types of gin, named as Geneva, Old Tom and Warrington. These were basically generic terms for sweet, medium and dry gins. Geneva or jenever was the continental type - I drank far too much of it one night in Rotterdam back in the '60s and have never touched it since! 'Old Tom' was traditional English pub fare and is rarely seen today. Closest to the modern 'London Dry' type of gin like Gordon's and Booth's would have been 'Warrington', which might also have been a generic type but certainly appears in the brand name for 'Greenall's Warrington Dry Gin', which was distilled in the town of that name and is still available today.
What I'm saying in a roundabout, rambling and possibly drunken way is that the wine lists are generally not very helpful in identifying specific brands, and it's quite possible that this was deliberate as supplies might have varied. But we do know that all three types of gin were offered at the same price of sixpence per glass, very pricey considering the 18th century tradition of "drunk for a penny, dead drunk for twopence".