Hi Tim,
One of the problems with the naming system for steam engine parts and functions is that it is really engineering shorthand.
The simple way I think to understand the difference is to look at the plain D Slide valve. The most common arrangement is three ports. The center port is always exhaust. The other ports that go to the top and bottom of the cylinder are called the steam ports but they only admit steam when the slide valve is not covering them. The steam line is connected to the steam chest which is on the outside of the valve. The steam port is actually the exhaust passage when the slide valve covers both the center exhaust port and the steam port. Full boiler pressure is on the valve stem packing. This is clasic outside admission.
You mentioned Peabody so Plate IX shows a typical piston valve with inside admission. The outside center of the spool valve is at full boiler pressure. The spool valve or piston valve is hollow and one end of the piston valve bore is connected to the exhaust line. The other end is the valve stem and packing gland. The ports in and out of the cylinder are still as always called the steam ports. In this case the central port is subject to steam pressure so It is known as inside admission.
The Docksteader valve gear files can be used to study the difference between inside and outside admission. Most of the common types of valve gear have a file that shows D slide valves and piston valves. If one of these are put in motion and the graphics and sound turned on the whole cycle can be viewed. The program changes color to indicate steam or exhaust. The exhaust chuff can be heard when the "steam port" is connected to the exhaust.
Piston valves can be outside admission. Check Plate IX in Peapody "Valve Gears for Steam Engines" simply change the "S" for steam and the "E" for exhaust and that is now outside admission and full boiler pressure is on the valve stem packing gland. The center port or passage is now connected to the exhaust line, and the center of the valve and the bore of the valve is connected to the steam line. As this just reversed the valve events by 180 degrees the eccentrics also have to be flipped 180 degrees for the proper valve timing.
Professor Cecil H. Peabody was the head of the Marine Engineering Dept. at M.I.T. He started the department and is one of the best authors in my opinion on steam engine valve gears.
I hope I am clearing things up and not confusing the issue more than nessary.
Cheers Dan