How literally should I take the Shimano compatibility tables? They seem OK about compatibility, but are not detailed enough about incompatibility.

I have a bike set up with 4500 kit running 2 x 9 gears which I want to convert to 3 x 9. I have a 4503 triple chainset, a 4603 LH shifter and a 2303 triple FD which I would like to use.

Will I have any compatibility problems? I don't find the answers I need in the compatibility tables.

With my older mountain bikes I just mix and match with no problems, but this is my first fairly modern road bike and I'm not sure if I'm over thinking it.

I've mixed 'n' matched Shimano parts across several decades plus combined non-Shimano with Shimano with (usually) no ill effects. Shimano's compatibility tables are usually very conservative and sales-oriented. The reality is that many of their parts are cross-compatible across their range aside from a few well-documented inconsistencies.

Short version: go for it. You'll be fine.

Double -> triple = longer chain, thus greater chainwrap capability of RD

Plan to use a RD with a longer cage (GS) if your current double has a short cage RD (SS)