Ruby, 322 bytes
->e,a,i,w,u,l,t{s='<svg viewBox="0-90 358 180"style="background:tan"> ' (2*628).times{|j|v=j*0.01 r=a*(1-e*e)/(1+e*Math.cos(v)) t+=r*r/(u*a*(1-e*e))**0.5*0.01 x,y=(1i**((v+w)/p=1i.arg)).rect q,z=(y*1i**(i/p)).rect s<<"<circle r='1'cx='#{(((-x-q*1i).arg-t/l*p*=4)%p*57).round(1)}'cy='#{(Math.asin(z)*-57).round(1)}'/> "} s} A function returning an SVG image.
First working version. Iterations are limited to 2 orbits and output data is rounded to prevent exceeding the byte limits for TIO.run and for text upload of sample output on this site. Rounding will be removed in the golfed version, as will newlines in the output.
I/O requirements
Arguments as follows
Orbit shape: e=eccentricity; a=semimajor axis Orbit orientation: i=inclination; w=argument of periapsis (in radians) Grav. parameter: u Planetary rotation: l=day length; t=start time Per the test cases in the question, there are no specific physical units for a and u, but the orbital period arising from them is such that it is consistent with the units of time used for l and t.
Function returns an SVG image with coordinate 0,0 at the centre. Tan background is used delimit the edge of the plot (Tan is one of only two SVG colours with only 3 letters, the other being red.) Calculations are done in radians but the output is crudely scaled by a factor of 57 (180/PI=57.2958 rounded down.) The width of the plot is therefore only 358 plot units. Calculated longitude values must be reduced MOD 2*pi to ensure they stay in the plot. This means the plot goes from 0 to 2*pi, but negative signs in -x-q*1i impart a 180 degree phase shift so that the calculated 0 longitude is in the centre of the plot. Latitude is +/-90 (limited to +/-89.5 plot units by the scale factor.) The aspect ratio is such that 1 degree is equal length in both longitude and latitude directions (not stretched horizontally like the plots in the question.)
The image is missing some whitespace required by the SVG spec for golfing reasons, but works fine on Chrome and Edge.
Explanation
Explanation and golfing to follow.
Sample output
Below is output for the first test case. It is recommended to hit run code snippet before expanding, to avoid getting a horizontally wide but vertically very short window requiring excessive vertical scrolling. It seems the plot is autosized based on the width only, so narrowing the browser window can help eliminate vertical scrolling.
<svg viewBox="0-90 358 180"style="background:tan"> <circle r='1'cx='9.9'cy='63.1'/> <circle r='1'cx='11.2'cy='63.1'/> <circle r='1'cx='12.4'cy='63.0'/> <circle r='1'cx='13.6'cy='63.0'/> <circle r='1'cx='14.9'cy='63.0'/> <circle r='1'cx='16.1'cy='62.9'/> <circle r='1'cx='17.3'cy='62.9'/> <circle r='1'cx='18.6'cy='62.8'/> <circle r='1'cx='19.8'cy='62.7'/> <circle r='1'cx='21.0'cy='62.6'/> <circle r='1'cx='22.2'cy='62.5'/> <circle r='1'cx='23.4'cy='62.4'/> <circle r='1'cx='24.6'cy='62.3'/> <circle r='1'cx='25.7'cy='62.1'/> <circle r='1'cx='26.9'cy='62.0'/> <circle r='1'cx='28.0'cy='61.8'/> <circle r='1'cx='29.2'cy='61.7'/> <circle r='1'cx='30.3'cy='61.5'/> <circle r='1'cx='31.4'cy='61.3'/> <circle r='1'cx='32.5'cy='61.1'/> <circle r='1'cx='33.6'cy='60.9'/> <circle r='1'cx='34.6'cy='60.7'/> <circle r='1'cx='35.7'cy='60.4'/> <circle r='1'cx='36.7'cy='60.2'/> <circle r='1'cx='37.7'cy='60.0'/> <circle r='1'cx='38.7'cy='59.7'/> <circle r='1'cx='39.7'cy='59.5'/> <circle r='1'cx='40.7'cy='59.2'/> <circle r='1'cx='41.6'cy='58.9'/> <circle r='1'cx='42.6'cy='58.7'/> <circle r='1'cx='43.5'cy='58.4'/> <circle r='1'cx='44.4'cy='58.1'/> <circle r='1'cx='45.3'cy='57.8'/> <circle r='1'cx='46.2'cy='57.5'/> <circle r='1'cx='47.0'cy='57.2'/> <circle r='1'cx='47.9'cy='56.8'/> <circle r='1'cx='48.7'cy='56.5'/> <circle r='1'cx='49.5'cy='56.2'/> <circle r='1'cx='50.3'cy='55.8'/> <circle r='1'cx='51.1'cy='55.5'/> <circle r='1'cx='51.9'cy='55.2'/> <circle r='1'cx='52.6'cy='54.8'/> <circle r='1'cx='53.4'cy='54.4'/> <circle r='1'cx='54.1'cy='54.1'/> <circle r='1'cx='54.8'cy='53.7'/> <circle r='1'cx='55.5'cy='53.3'/> <circle r='1'cx='56.2'cy='53.0'/> <circle r='1'cx='56.9'cy='52.6'/> <circle r='1'cx='57.5'cy='52.2'/> <circle r='1'cx='58.2'cy='51.8'/> <circle r='1'cx='58.8'cy='51.4'/> <circle r='1'cx='59.4'cy='51.0'/> <circle r='1'cx='60.0'cy='50.6'/> <circle r='1'cx='60.7'cy='50.2'/> <circle r='1'cx='61.2'cy='49.8'/> <circle r='1'cx='61.8'cy='49.4'/> <circle r='1'cx='62.4'cy='49.0'/> <circle r='1'cx='63.0'cy='48.6'/> <circle r='1'cx='63.5'cy='48.2'/> <circle r='1'cx='64.0'cy='47.7'/> <circle r='1'cx='64.6'cy='47.3'/> <circle r='1'cx='65.1'cy='46.9'/> <circle r='1'cx='65.6'cy='46.5'/> <circle r='1'cx='66.1'cy='46.0'/> <circle r='1'cx='66.6'cy='45.6'/> <circle r='1'cx='67.1'cy='45.1'/> <circle r='1'cx='67.6'cy='44.7'/> <circle r='1'cx='68.0'cy='44.3'/> <circle r='1'cx='68.5'cy='43.8'/> <circle r='1'cx='69.0'cy='43.4'/> <circle r='1'cx='69.4'cy='42.9'/> <circle r='1'cx='69.9'cy='42.5'/> <circle r='1'cx='70.3'cy='42.0'/> <circle r='1'cx='70.7'cy='41.6'/> <circle r='1'cx='71.1'cy='41.1'/> <circle r='1'cx='71.5'cy='40.7'/> <circle r='1'cx='71.9'cy='40.2'/> <circle r='1'cx='72.3'cy='39.7'/> <circle r='1'cx='72.7'cy='39.3'/> <circle r='1'cx='73.1'cy='38.8'/> <circle r='1'cx='73.5'cy='38.3'/> <circle r='1'cx='73.9'cy='37.9'/> <circle r='1'cx='74.3'cy='37.4'/> <circle r='1'cx='74.6'cy='36.9'/> <circle r='1'cx='75.0'cy='36.5'/> <circle 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