I think it depends on context. My wife and I have sailed together so much, and we fully split duties. Whoever is at the wheel releases the jib sheet, and the other grinds. The only conversation is something like, I think I'll tack now. The other person goes over to the winch as says OK. Then the helms person turns the wheel. When we have guests, no need to interrupt the conversation, unless they want to play. If they do, I usually start them on the wheel, it's easier, then if they are interested, teach them to grind. No formal language, more like "ready?" and taking a look to see the wrapped the sheet the right way, etc.
When we talk about lines, we do use the right terms. When we talk about direction we do use port and starboard. But we don't need to talk much, too many years.
And let me strongly agree with capta. ABSOLUTELY no words spoken when anchoring. We stay in the cockpit and discuss the plan quietly and perhaps motor by the spot. I casually walk forward, she maneuvers back to the "spot" and brings the boat to a halt. I lower the anchor. Hand signals only including pointing where to go, fingers for RPMs in 100's in reverse to gradually set, closed fist for neutral, etc. All signals given casually, particular singular fingers avoided
. When set and snubbed, walk back to the cockpit and enjoy the show over cocktails as others arrive.
It's not how you anchor, it's how you look (and sound). Same thing approaching docks. Plan first, vocalize in the rare occasion that you need to change the plan.
That said, I suppose in a school situation, or a yacht club or race when the crews change a lot, etc, common language might be helpful.