This gets onto very shaky ground. What if I'm in a console and only browse with w3m or lynx? What if I simply disable all Images and Javascript (older web browsers made disabling images really simple; back when we had very limited dial-up bandwidth).

Can websites dictate what web browsers you use? Are you violating an agreement by using a different User-Agent?

No, no and no. You are requesting data and your browser renders it. If you write your own browser to request and render the data differently, I don't see any issue with copyright.