Is there any difference between a Cat 5e and a Cat 6 RJ45 plug?

Question

I described this issue here Cat 6 cable failure where I found that my Cat 6 buriable, waterproof cable (over 200' long) had been severed by a rodent.

Cat6 connectors worked for data transfer but Cat 5e connectors did not for a Cat 6 cable that tested fine with testers with either type plugs attached.

I got Ideal Industries brand Cat 5e RJ45 plugs, like these: https://www.lowes.com/pd/IDEAL-25-Pack-Rj45-Data-Cable/1002793 (8-position, 8-contact RJ-45 modular plug for Cat 5e, 100Ω balanced twisted pair 100 MHz cables) and a Cat 6 connector to connect the two pieces of Cat 6 cable, and attached the plugs to each end of the Cat 6 cable, and at the break, connected everything, everything tested fine continuity across all 8 wires, using both one of those flashing LED cheap testers, and also with a Southwire M400TP digital tester, which amounts to the same thing I believe, just with a digital screen: http://www.southwiretools.com/tools/m/tools/M400TP

Although everything tested fine, still I couldn't get data transfer between the two routers - the Cat 6 cables plugged in at each end would not turn the little LED light green to indicate data transfer, and the routers never recognized each other.

Then I went I got these Ideal Industries Cat 6 RJ45 solid plugs: https://www.lowes.com/pd/IDEAL-25-Pack-Rj45-Data-Cable/1000331509 (Single piece Cat 6 connector same preparation as Cat 5e connector Pair twists maintained within 1/4 in. -3/8 in. of pin contact) and used the same Cat 6 connector to connect the two pieces of Cat 6 cable, and replaced every plug with these, ran the tests again, everything fine, and now the data transmits and works fine.

Why would the Cat 6 plugs work any different from the other? I am running under 300 Mbps speed.

I suppose a secondary Q is, why would the cables test fine with the Cat 5e plugs, with the testers and yet not work for data?