你可以跟他们说：我想理发或者我想剪一个新的发型或者我想剪头发

Both of them they are right. But the point is the first exactly means u wanna cut ur hair. The second one means u wanna fix ur hair. Both of them actually have the same meaning lmao.

Simply put 1.when it comes to cutting hair，these two words share the same meaning. 2.剪informal/理formal But，nowadays in China，it makes no different between 剪发 and 理发.So choose whatever u like. 3.related words u have to keep in mind 理发店 / 造型师 /发型 /染发 /烫发 /洗剪吹

剪头 is colloquial and should be only used in spoken language while 理发 is much more formal. The barber's in Chinese is 理发店 rather than 剪头店. The verb 剪 means to cut sth. into two pieces (but 剪头 does not mean to cut your head into two pieces, lol). However, the verb 理 means cut (hair) only when it's in the fixed word 理发. It has various of meanings in other combinations. If you want to tell your barber you want a haircut, you can say 我想剪个头 or 我想理个发, both are ok.

In Taiwan they mean the same. But usually we use "理"頭髮 on man, while "剪"頭髮 on both man and woman. EX : 去把頭髮理一理 (Go cut your hair) . I think you can simply say 我想剪頭髮 (I wanna cut my hair) or 我想換髮型 (I wanna change my hair-style)