For what you describe, I would not use the Animation Presets, but instead Keyframe the fixed Effect>Motion>Position. With Keyframes, you can set the exact speed, and Velocity, of the animation, plus you can pause it, where ever, and for however long you wish.
To create a Title, based on an existing one, but in no way linked to the other (remember, when you drag a Title to the Timeline, you are ONLY dragging an Instance of that original Title, so that any change will affect all Instances), you will want to Rt-click on that first Title, and choose Duplicate. This will create a new Title, not just an Instance of the original. You can change that Duplicate (I would Rename it), as you wish, without affecting the original. [In PrPro, there is a little icon, right at the top of the Titler, to do just that.]
Good luck, and hope that I got what you needed correct.
Bill, thanks for the help.
I'm using the "DropInByCharacter" Text Animation for this effect. While it certainly is possible to do this with keyframes, I think it would take forever!
I tried "duplicating" the title by clicking the "Organize" tab, then "Project" button. Then I right-clicked on the animated title and chose "Duplicate." Then I pasted it into the timeline. Then I double-clicked the new title, selected the animated text, opened the Text Animation pane (under the fonts), and clicked "Remove." Unfortunately, this once again removed animation from BOTH of the titles, so they are still linked.
I solved the problem by just copying the text object from the animated title into a new titled I created. I would still be interested to know how to "unlink" titles.