Different terms have arisen in different contexts to achieve different purposes.
It's unfortunate when IRA owners get so muddled trying to sort out IRA terminology they lose sight of the really important stuff. This "dictionary" will attempt to clear up the confusion, so IRA owners can get back to focusing on the big things, like taking required distributions, naming a beneficiary, and avoiding mistakes in the rollover/transfer process.
How did this stuff get so confusing? Because different terms arose in different contexts to achieve different purposes. Supposedly everything about the individual retirement account comes directly from the Internal Revenue Code, but in the real world, terms evolved to achieve the purposes of the IRS (to regulate IRAs) and the IRA providers (to administer and market them).
"Traditional" IRA: The Code never uses the term "traditional IRA," so technically there is no such thing. There are only "IRAs" (individual retirement accounts), see § 408, and "Roth IRAs" (§ 408A). But when Roth IRAs were added to the Code (in 1998) it became necessary to distinguish between the two types of IRAs. So the Treasury, in its regulations dealing with Roth IRAs, started using the term "traditional IRA" to mean an IRA that is not a Roth IRA. See Treas. Reg. § 1.408A-1 et seq. The term "traditional IRA" is now in common usage to mean a non-Roth IRA.
"Rollover" vs. "Contributory" IRA: These terms are never used in the Code or the Regulations, but IRA providers sometimes use them.
Back in ancient times (before 1993), the Code made a distinction between an IRA that contained nothing other than rollovers from one or more qualified retirement plans vs. an IRA that contained money contributed directly by the participant. Funds from a "pure" rollover IRA (then called a "conduit IRA") could be rolled back into a qualified plan, whereas no money distributed from an IRA that contained participant contributions could be rolled into a qualified plan. That Code distinction is gone (pre-tax funds from an IRA can be rolled into a qualified plan, regardless of the source of contributions to that IRA), and there is now no known tax reason for making a distinction between contributory IRAs and "pure" rollover IRAs.
However, for asset-protection purposes, there still is a difference--federal bankruptcy law grants a total exemption for IRAs that contain nothing other than rollovers from qualified plans, whereas the exemption is limited to $1 million (adjusted for inflation) for other IRAs. Whether for that reason or some other unknown administrative reason, some IRA providers tend to label every IRA account they administer as either "rollover" or "contributory." Since these labels have no legal or tax significance, this IRA provider policy is not of significant concern. While it's nice if the IRA provider uses the correct label for your account, the label does not change reality: Your IRA is either "contributory" or "pure rollover" (for bankruptcy purposes, if you end up in bankruptcy court) based on what contributions have actually funded it, not the label the IRA provider attaches to it.
"Self-directed" IRA: A self-directed IRA is one for which the IRA owner himself (the participant) makes the investment decisions. Almost every IRA is a self-directed IRA. It is unusual to encounter an IRA where investment decisions are made by someone other than the participant. In other words, the IRA that is not "self-directed" is actually rather rare.
Unfortunately this term has been usurped by the investment industry to apply to IRAs that are invested in assets other than cash, CDs, mutual funds, and publicly traded stocks and bonds. Promoters of arrangements for IRAs to invest directly in real estate, private businesses, hedge funds, and other "non-vanilla" investments call their products "self-directed IRAs." This meaning has become so common that the owner of a plain ordinary IRA invested in mutual funds can get upset if he's told he has a "self-directed IRA."