It seems that what you are seeing as a 1M resistor is actually the NTC thermistor (aka Brimistor which was a trademark of STC). Its function is to slightly delay the operation of RN to ensure a full open transition. Bypassing it, as suggested in note 1 for lower power motors, simply causes RN to operate immediately. If the starter works with it bypassed, the Brimistor itself is probably faulty.
OK, you asked.
Before we start:
a) Series-parallel starting refers only to the connection of the run / main winding. It is the equivalent of star-delta starting but for single-phase, in that it lowers the voltage across each winding to reduce the starting current until up to speed, without the use of resistors or transformers. The start winding configuration and capacitor arrangement is a separate consideration.
b) There are two aspects to the start-run transition; switching the two run windings A1-A2 and A3-A4 from series to parallel, and disconnecting the start capacitor.
c) There are two possible capacitor configurations; capacitor-start induction-run, in which the capacitor winding is completely disconnected in run, and capacitor-start capacitor-run, in which the winding is disconnected from the start capacitor but remains in circuit via the run capacitor.
d) There are two possible centrifugal switch configurations; either it switches the start capacitor directly (in which case the starter timer is only responsible for the series-parallel transition) or it controls the ST contactor, in which case that contactor controls both the series-parallel connection and the start capacitor.
e) ST4 and RN2 are mutual interlocks that prevent ST and RN operating at once, which would short the main circuit L-N.
Pre-conditions for starting:
a) C2, RN2 and TM2 are all closed, i.e. the starter is not already in either the starting or running state and the timer is reset.
b) If the centrifugal switch is in the control circuit, it must be closed, i.e. the motor is not at speed.
START BUTTON PRESSED
Start button energises ST and TM
ST OPERATES. TM BEGINS TIMING
ST2 prepares a holding circuit for ST and TM, ready for when C2 opens
ST3 energises C
ST4 opens to prevent RN operating
ST main contacts 3-4 and 5-6 connect run winding terminals A2-A3 for series starting
ST main contact 1-2 connects start capacitor circuit (if controlled by starter and not directly by centrifugal switch)
C OPERATES
C2 opens to leave ST held only on ST2
C3 prepares a holding circuit for C, ready for when ST3 opens
C main contacts connect motor to mains
MOTOR RUNS UP TO SPEED
Motor run windings are in series, capacitor winding is in series with start capacitor (if CSIR) or start and run capacitors in parallel (if CSCR)]
TM TIMES OUT (OR CENTRIFUGAL SWITCH OPENS IF IN CONTROL CIRCUIT) WHICHEVER IS FIRST
TM2 or centrifugal switch disconnects ST
ST RELEASES
ST2 opens ST's holding circuit and disconnects TM.
ST3 opens, leaving C held only on C3
ST4 closes to energise RN via Brimistor.
ST main contacts disconnect run winding series connection and start capacitor.
BRIMISTOR HEATS UP
Brimistor resistance falls in a second or so, to the point where it is low enough for RN to operate.
RN OPERATES
RN2 opens to lock out ST
RN3 closes to bypass Brimistor
RN main contacts reconnect run windings in parallel
IF CENTRIFUGAL SWITCH IN MAIN CIRCUIT, IT OPENS
Start capacitor disconnected
Motor is now in running state.
No TL;DR option here. Either you follow it or you don't!
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Re. your specific motor, it's hard to tell from the pics. What I think I am seeing is that the two capacitors are in parallel, making it CSIR, and that there is no centrifugal switch. Therefore, the start winding is controlled by ST and disconnected under timer control at the same time as the series-parallel transition. The maximum timer setting is limited by the start capacitor duty cycle as noted in the instructions.