Most G.R.textbooks introduce time as an extra dimension... So although I can not mentally imagine this, I think of it as an extra line, "orthogonal" to the 3 normal space dimensions.
Don't. It's a dimension in the sense of measure, not in the sense of freedom of motion: I can hop forward a metre but you can't hop forward a second. Think about what a clock really does. It doesn't literally measure the flow of time like some kind of cosmic gas meter. It features some kind of regular cyclical internal "local" motion, be it of cogs or a crystal, and it shows you a cumulative result that you call the time. Then think about what we're really doing with all this. We're modelling motion through space "over time". Einstein gave us the equations of motion. That extra dimension is there because ours is a world of space and motion. When you look up at the sky you don't see worldlines and light cones, you see planes and meteors and planets and stars and light, and everything is moving. Have a look at A World without Time: The Forgotten Legacy of Godel and Einstein. It's like time exists like heat exists. Temperature is an average measure of motion, time is a cumulative measure of motion, and just as you can't literally climb to a higher temperature, you can't literally travel to another time. Because you can't move through a measure of motion.
Then the text books introduce the equivalence principle
The equivalence principle was an "enabling" principle, wherein Einstein's happiest moment was when he realised the falling man was not subject to a force. However he said this in 1916:
"This is by no means true for all gravitational fields, but only for those of quite special form. It is, for instance, impossible to choose a body of reference such that, as judged from it, the gravitational field of the earth (in its entirety) vanishes".
That's because the equivalence principle applies to an infinitesimal region only. So it doesn't apply to the room you're in. Being in a real (="quite special") gravitational field is like accelerating through space, but it isn't exactly the same.
and use it, involving, if I understand it correctly, purely the time direction, to illustrate the deflection by the Sun of say, a photon from a distant object
Light curves because the speed of light varies with position. Search the Einstein digital papers for Einstein saying that. Also see Shapiro's 4th test of General Relativity: "the speed of a light wave depends on the strength of the gravitational potential along its path". You can liken this variable coordinate speed of light to a "tilt" in your time dimension, or "spacetime tilt", see this article for a mention of tilted light cones. But as far as I know the spatial element to the Shapiro delay contributes only a tenth of the total. As far as I know light is deflected twice as much as matter because of the wave nature of matter.
and they state that, without also allowing for the curvature of space near a massive object, we will measure only half of the actual deflection observed.
Can you give me a reference? It might be an idea if you asked a dedicated question on this.
So can I take it that my naive idea of a "time" as simply a linear "direction" is actually equivalent to a volume of space, as the deflection of a photon through time is 1/2 of it's actual total deflection?
I'm sorry, but what you're saying doesn't sound right.
If my assumption is incorrect, could somebody explain more regarding the subtleties involved in the physical intuition behind 4-D spacetime.
4D spacetime is a mathematical model for motion through space over time, starting with the motion of light. IMHO it's easier than you think.