We study the online facility location problem with uniform facility costs in the random-order model. Meyerson's algorithm [FOCS'01] is arguably the most natural and simple online algorithm for the problem with several advantages and appealing properties. Its analysis in the random-order model is one of the cornerstones of random-order analysis beyond the secretary problem. Meyerson's algorithm was shown to be (asymptotically) optimal in the standard worst-case adversarial-order model and $8$-competitive in the random order model. While this bound in the random-order model is the long-standing state-of-the-art, it is not known to be tight, and the true competitive-ratio of Meyerson's algorithm remained an open question for more than two decades. We resolve this question and prove tight bounds on the competitive-ratio of Meyerson's algorithm in the random-order model, showing that it is exactly $4$-competitive. Following our tight analysis, we introduce a generic parameterized version of Meyerson's algorithm that retains all the advantages of the original version. We show that the best algorithm in this family is exactly $3$-competitive. On the other hand, we show that no online algorithm for this problem can achieve a competitive-ratio better than $2$. Finally, we prove that the algorithms in this family are robust to partial adversarial arrival orders.