It doesn't appear that a lot of research has been done on this topic, but the research that has been conducted seems fairly clear. The Wikipedia article on lead-glass safety contains a brief description of the safety aspects.
Going just a little deeper, I found two abstracts. One deals with short term use, the other with long term storage.
As the article you linked mentions, long term storage of acidic or alcoholic beverages is not recommended. This Lancet abstract describes an experiment which found significant increases in lead concentration long term storage. Concentrations rose from 79 ug/L initially to 3817 ug/L after four months, and up to 21,530 ug/L after "a long time." For comparison, a typical diet will contain around 70-80 ug of lead a day.
However, short term use of lead-glass containers showed relatively small amounts of lead-leaching. According to this PubMed abstract, under "conditions that are likely to occur to a consumer" the worst case lead leaching amounted to only 14.5 ug in a 350 mL (12 fl. oz.) soda or about 41 ug/L. The researchers concluded that the lead increase would not significantly affect health.
On the other hand, the US CDC "Action Level" for lead in water is 15 parts per billion, or about 15 ug/L. While the soda case (worst case) from the PubMed study exceeds this level, one presumes that such consuption is not a regular occurrence. Regular use of a lead-glass beverage container would probably be contraindicated.