There are three drinks you can call a martini:

• A martini is gin and vermouth, maybe with some bitters if you like
• A vodka martini is vodka and vermouth, bitters again optional
• A vesper martini is gin, vodka, and lillet blanc
• Any of the above can be made “dirty” with olive brine if you want

Anything else is a cocktail in a martini glass. No shade if you like apple schnapps, lemon juice, and vodka, drink what you like, but it’s not a martini.

Ritual can be entirely divorced from theism or a belief in the supernatural. Nothing wrong with creating a secular ritual if it helps you in some way.