What ever is feeding the intial old lite you are removing,that circuit should be enough to add 3-4 pot lites.
Your on the right track,remove the old lite and cut that old circuit into you first new pot lite,just carry on to the others from there, 14/2 is all you need.

Where did I get the idea that I needed 14/3? I thought that was required in order to wire the lights in parallel; otherwise with 14/2 they'd be in series and dim. No?

No. Just use 14/2. At each light fixture, connect the blacks from both the "in" and "out" lines and the black from the light fixture all together. Same for whites and grounds. They aren't in series unless the current needs to flow through one of the lights to get to the next. By connecting all the blacks together, the current flows through the wires to get to lights 2-4, not through light 1.

Sounds simple enough. So just 14/2 all the way around, and the loop ends at the fourth light?

In what scenario(s) would you use 14/3?

The most common uses for 14/3 in a home would be wiring a light to 2 separate switches or wiring a fixture that does 2 things - like fan/light combo.

But I thought I once saw a posting here where someone wired two outdoor garage lights and they were dim. He was told to run 14/3 between the lights, and that fixed it. Am I confused?

If the power came to the first light and the switch was at the end of the run... 14/3 would be required between the lights. I believe that was the case on the post you refer to.

please see How to Wire a Circuit "In Parallel"

Here is a link that might be useful: How to Wire a Circuit

I believe I have a switch loop in this situation. There is only one cable in the switch box, with the white one marked black. Would this change my cable needs if I wanted to add these recess fixtures? Is 14/3 necessary now? Thanks.

White WIRE is marked black, that is.

At the existing fixture remove the white and black from the fixture only and continue on as sparky outlined. Black and white to each fixture.

There will be more wires in the box but don't mess with them, they are feeding the switch.

Get a wiring book, that's what I did, and look at examples of wiring. Things will be a lot clearer by looking at pictures.

OK, here's my next question then. I have some concerns about the wood in my house (probably unfounded). What are my options for securing the cable above the suspended ceiling *without* drilling the joists? Aside from just stapling to the bottom edge of the joist (the worst options?) can I use a light PVC conduit and secure that? Thanks.

If you have room above the ceiling just run some furring strips, 1x2 or 1x3, and staple the wire cable to that.

How do you mean? Attached perpendicular to the bottom edge of the joists?

It's a suspended ceiling, so yes, I have maybe a foot of height to the subfloor above.