Asking what kind of customer someone is, is very important to you, but in UX design you need to think about what is important to your users. Focus on your offerings, your users have an idea of what they're looking for, they're just trying to figure out if you're offering it.
The "Who are you?" model is off-putting to users because it creates pressure and uncertainty of which category they fit into. I think you already know that, since you included a "?" for those who don't know. So go with your gut, this isn't a good idea! Think about whether it is really critical for you to create different experiences for different people. Are they really that different to warrant a different site experience? How else might you be able to collect information on what people visiting your site care about?
Rather than a potentially confrontational, "Who are you?" you can organize your site by what you offer. I see that below with your "Projects" section. That's the right idea and this is where you can create segmentation based on what your customers care about.
A helpful trick is to use real world scenarios, because they are very powerful. Imagine walking into a store, but before you went in, someone was standing there with a clipboard asking you "So are you looking for something specific, a whole new outfit, browsing, or going to the sale rack?" Yikes! You'd run in the other direction. A better experience would be to let a user walk in freely, let them notice the layout of the store and observe that they went to the sales rack. You can make helpful suggestions once you see where they're looking, "You know anything you buy that's on sale is another 10% off today." Relaxed and helpful when needed, that's how you want your site to feel when you present your services. So go with categorizing your offerings in ways that might reflect what kind of customer they might be and meet them where they are once they get there.
On this note, you can try to include action items in contextual ways to drive users toward contacting you with inquiries. You might think about including a "Like what you see? Send us an inquiry!" link at the bottom of each project page. :) Drive, drive, drive. In context, in context, in context.
Finally, once they get to the inquiry form, you could do something very direct to get the data you seek. Include a drop down menu item asking the customer who they are. :) You never know what kind of customers you might be attracting so let them tell you who they are, but do this in the form-like setting of the inquiry itself, where they user will expect it, not in the design of the site.