TM4C123GE6PZ: why waveform for XOSC1 pin is not pure sine-wave

How sure are you - and the client - that, 'Their act of measurement' has (not) colored their findings?     Scope probes do (often) introduce  circuit intrusion effects  - is that not so?

Might you measure & present a similar scope cap - made upon a (near) identical 'XOsc' pin - from that same scope & probe - yet captured from (another source) - which presents a 'perfect sine wave?'

Without that comparative presentation -  client's technique - may require (some) adjustment...

You should be able to quickly produce a 'Frequency Scaled replica' of the System Clock - generated via an MCU's Timer pin (Set to PWM Output Mode) - which can confirm the 'proper' operation of the System Clock Generating circuitry.    

Such intrusive, 'direct measure' - as you've shown - (may) cause and/or add distortion - not present when the probe is removed...     Use of the 'Scaled Frequency Technique' - which AVOIDS all potential 'XOsc' disturbance - is a 'known & proven' method to insure & confirm - System Clock performance...    (in fact - certain of our clients - configure & allow such Timer Output to remain - to be regularly measured - by their normal operating software...)

Below is the waveform for the cystal another terminal XOSC0 pin which captured by same scope & probe in same board, it can say a perfect sine wave. So I think the scope & probe & measure method should be ok.  

I am suspect if the distortion for XOSC1 pin waveform is TM4C123 default behavior? If yes, what reason to design such waveform? If no, what possible reason maybe trigger such waveform?

Below is the waveform for the cystal another terminal XOSC0 pin which captured by same scope & probe in same board, it can say a perfect sine wave. So I think the scope & probe & measure method should be ok.  

I am suspect if the distortion for XOSC1 pin waveform is TM4C123 default behavior? If yes, what reason to design such waveform? If no, what possible reason maybe trigger such waveform?

Well done - both the frequency & signal range are 'consistent' - thus your scope/probe method - employed upon this particular (most recent) Xosc - prove acceptable.

How common are the crystals (assumed xtals) and bypass caps - across (both) your measured devices?    Different xtal vendors may specify different loading conditions - that deserves (some) review.

That waveform distortion (may) suggest an inadequate power condition - forced upon the MCU.     Are all  Vdd bypass caps emplaced - and have: Vdd, VddA, and (especially) Vddc all been measured & confirmed?

As always - upon 'how many'  such boards does this effect occur?     (you note two - a broader testing would be more informative)

Yet again - unless vendor here over-rides - my belief is that the, 'Timer - commanded into PWM, Frequency Replica (divided) Output Mode' - provides the best 'feedback' - as to the 'success' of the external xtal circuit.