As you can see, there are differences not only in the ranked border with another league, but also divisions within the league. For example, a Silver 5 border will have a slight difference compared to the Silver 1 border. This difference was intended to push players to advance in rank queue even if they were not able to advance into the next league.
Depending on which queue you earned your rank in (normal rank or ranked teams), these ranked borders will only be displayed in that specific queue. This means your rank border earned in solo or dynamic queue will not be displayed in a ranked 5 teams queue and visa versa.
Silver 5 to Silver 1
Gold 5 to Gold 1
Platinium 5 to Platinium 1