Cleaning of Digital Crown

The first Watch is water resistant but not recommended for submerging.

Apple Watch Series 1 and Apple Watch (1st generation) are splash and water resistant, but submerging Apple Watch Series 1 and Apple Watch (1st generation) isn't recommended.

Apple Watch Series 1 and Apple Watch (1st generation) have a water resistance rating of IPX7 under IEC standard 60529. Apple Watch Series 2, Apple Watch Series 3, and Apple Watch Series 4 have a water resistance rating of 50 meters under ISO standard 22810:2010. Classic Buckle, Leather Loop, Modern Buckle, Milanese, and Link Bracelet Bands aren't water resistant.

https://support.apple.com/en-gb/HT205000

Read this to help you

https://support.apple.com/en-gb/HT204522

You need advice direct from Apple's support team, use link Contact Support

Not sure why they need to contact Apple support? Apple own advice to clean it has been post?

As the OP pointed out, Series 1 does not claim water resistance. Rinsing with water as described for later versions could be a problem, so the safe option is to ask the experts.