When talking to American clients, should I say "smoothie" or "milkshake"?

A milkshake usually contains only milk and ice cream. However, sometimes strawberries or bananas can be added.

If the beverage is mostly fruit and ice, then I would call it a smoothie. If you add some milk, nuts, or ice cream to it I would still call it a smoothie.

The key ingredients determine the name. So if it is mostly ice cream and milk, it's a milkshake. If it's mostly fruit and ice then it's a smoothie. Also, milkshakes traditionally don't have nuts, protein powder, vitamin powders, or other additives.

In your example, I would call it a smoothie. If I saw smoothie on the menu I would expect the ingredients that you listed. Milkshake confuses things a bit. Stick to one.

For most Americans, a milkshake is served as dessert (usually at the end of the meal), although it may also replace the main beverage for the meal if one is feeling indulgent. Milkshakes are generally not associated with breakfast.

Smoothies, however, are generally viewed as a meal-replacement, most often for breakfast (when one is in a hurry to get to work), so it will generally contain protein, fiber, fruit, and enough ice and/or milk (or milk-like substitute such as yogurt) to blend all ingredients into a drinkable liquid.

One additional distinction is consistency. Milkshakes tend to be thicker than smoothies.

The critical thing I would go with is to make it clear that the smoothies contain milk and nuts. It's not a given that a smoothie will contain either (in the UK, most contain yoghurt as the base ingredient, or are pure fruit), and if someone is lactose intolerant or allergic, it can range from embarrassing to catastrophic to get that wrong.

For preference with British English, I would go with Smoothie; it's certainly more expected as a healthy breakfast drink, whereas milkshake is generally used in the context of a dessert or as an accompaniment to a meal (e.g. McDonalds).

If it's a lassi (which I know commonly are fruit, and sometimes use the pistachio nut or almonds) call it a lassi

An American going on an assignment in India should understand "lassi" as a dairy-and-fruit beverage at a baseline and may be pleased by menu verisimilitude as a recipe of Indian origin.