A couple notes...
For an externally inserted vulcanized tire plug, rubber cement is not required. It doesn't aid in sealing though it does act as a lubricant which can reduce the force needed to insert the plug. So pulling the plug strip halway into the tool is perfectly fine, if not ideal (greater range for the doubled plug to seat throughout the tire depth). If you don't have any cement don't worry, just use the reaming tool to make sure the hole is opened up a little. Always use the reaming tool anyway to clean out the hole. Keeping air in the tire while plugging is more about ensuring the tire is not deflecting a lot when you are attempting to push the plug in. Without the resistance the inflated tire provides, it is much harder work, and for lower profile tires you can bottom out the tool on the rim if the tire collapses inward. If you notice any air leaking after the repair attempt, it is most likely the puncture was at an angle and not perpendicular to the tread. In that case you may need to insert another plug that lines up better with the hole. But you need to remove the wheel to have any chance to inserting a plug at some angle other than 90 deg.