Waterloo Region has an obsession with roads changing names at intersections and it's definitely something that people who move here find to be extremely strange. A lot of it has to do with how the cities grew into each other and how the subdivisions were made so in this particular case Eagle St "ends" at Hespeler Rd and Pinebush Rd "starts" even though they're really the same road, one issue with calling the station Eagle would be people wouldn't necessarily know which end of Eagle St, is it the Preston end or the Hespeler end? This is why the stations are called Preston and then Pinebush. A similar issue arose with phase 1 when they were originally coming up with names as King was an idea that was floated but because of the line running along King it's arbitrary so they got rid of it. It's also why there is no Hespeler station since the LRT runs on Hespeler from Pinebush to the Delta.
The Delta is another strange example of this naming issue as you have Hespeler Rd turning into Water St and you also have Coronation Blvd turning into Dundas St all at the same intersection. However this is certainly not a problem solely in Cambridge, Kitchener and Waterloo both have some relatively major roads changing names constantly for example in Kitchener Lackner Blvd turns into Bingemans Center Dr which turns into Shirley Ave, which then turns into Wellington St, and then at both ends turn into local roads (Walter St, Glasgow St, Lackner Cres, and Idle Creek Dr), you also have Queen St turning into Queens Blvd, Park St turning into Jubilee Dr turning into Courtland Ave turning into Fairway Rd which eventually turns into Kossuth Rd. In Waterloo you have Columbia turning into Lexington, you also have Erbsville Rd turning into Ira Needles Blvd.
There's also a couple dozen other examples of this happening in KWC as well but on your smaller suburban local roads, for example Westforest Trail turns into Eastforest Trail which turns into Westheights Dr which then turns back into Westforest Trail. Commonwealth St turns into Abram Clemens Way, Donnenworth Dr turning into Amand Dr, and the list goes on.