couple things here:
1. Both API 600 and API 603 are gate valve standards. technically a globe valve will never meet either standard. That said what most people quote is globe valves that meet the applicable parts of API 603 and 600
2.For the most part API 603 and API 600 are similar however The biggest difference is wall thickness. API 600 is written for steel valves and has an allowance for corrosion on the steel. API 603 is written for corrosion resistant materials ( like stainless steel) and does not have the extra wall thickness allowance for corrosion.
3. Recently API released API 623 which is a new standard for steel globe valve equal to API 600 (for gate valves). Today is you want a quote for an API 600 style steel globe valve you should probably specify API 623 instead since the spec is written around the considerations of globe valve designs.
4. As of today there is not and API standard for globe valves equal to API 603, Mainly because the use of stainless steel globe valves in the petroleum industry is not high so they haven't had many requests for to write it.