Whether to have 2 or 3 alternative logos for your website ultimately depends on the specific goals and needs of your business.
Having multiple logos can give you more flexibility and the ability to switch up your visual identity based on different campaigns, seasons, or events. It can also help you appeal to different target audiences or convey different aspects of your brand.
However, having too many logos can be confusing for your customers and dilute the recognition and memorability of your brand.
Sticking with one logo can help establish brand consistency and recognition, making it easier for customers to remember and identify your business.
Ultimately, the decision should be based on your overall brand strategy and the image you want to convey to your audience. Consider working with your designer to create a strong primary logo, and then a few secondary options that can be used in different contexts as needed.