Couple thoughts.
One - A bad connection in thermostat. This may be internal and not on the terminal strip. Although I would make sure the terminals/wires are making good connection.

Two - The spliced wire to the outdoor unit. If you have a volt-ohm meter. Check the resistance of the wires to the outdoor unit. Secure power (Inside and outside) and connect the wires together on the outdoor unit. Then disconnect at the other end and measure the ohm reading. It should be low 0-1 to maybe 2 ohms resistance. If much higher, you will need a new pair of wires or find and fix the bad connections.

Three - There may be a bad connection to the medium speed winding. If you have a push together connector for the indoor fan motor wires, separate and check the male/female connections for clean and good connections. The medium winding on the motor may be failing,

Try the simple easy things first.
Ron