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The problem is that both titration curves are wrong: at the first equivalence points, the pH does not equal $\mathrm{p}K_\mathrm{a1}$ and the pOH does not equal $\mathrm{p}K_\mathrm{b1}$. As well, the two provided dissociation constants for carbonate ion, in the OP's upper figure, are also wrong.

The figure below shows two titration curves: 1) for 10 mL of 0.1 M "carbonic acid", with 0.1 M NaOH titrant (black rising curve) and 2) for 10 mL of 0.1 M sodium carbonate solution, with 0.1 M HCl titrant (red falling curve):

The pH of the starting 0.1 M carbonic acid is 3.67 and the pH of the starting 0.1 M sodium carbonate is 11.65. At the first equivalence point, for both titration curves, the pH is approximately 8.34. The $\mathrm{p}K$ values are as shown in the figure and are from (or derived from) the reference at the bottom.

The next two figures show Excel spreadsheet pH calculations. The first one shows that pH = 8.338 at the first equivalence point, for the neutralization of sodium carbonate solution with HCl:

At the first equivalence point, the solution is 0.05 M sodium bicarbonate plus 0.05 M NaCl. Ignoring ionic strength effects, the NaCl does nothing to the pH.

The next one shows that pH = 8.339 at the first equivalence point, for the neutralization of carbonic acid with NaOH:

At the first equivalence point, the solution is 0.05 M sodium bicarbonate. In both cases, the pH is approximately 8.34 and is consistent with this answer.

The two $\mathrm{p}K_\mathrm{a}$ values for carbonic acid are from D.G. Harris, Quantitative Chemical Analysis, 7th Ed., Appendix G, W.H. Freeman and Company, ©2007. The two $\mathrm{p}K_\mathrm{b}$ values for carbonate ion are obtained from the two $\mathrm{p}K_\mathrm{a}$ values for carbonic acid via $\mathrm{p}K_\mathrm{b1}$ = 14.00 - $\mathrm{p}K_\mathrm{a2}$ and $\mathrm{p}K_\mathrm{b2}$ = 14.00 - $\mathrm{p}K_\mathrm{a1}$.

The problem is that both titration curves are wrong: at the first equivalence points, the $\mathrm{pH}$ does not equal $\mathrm{p}K_\mathrm{a1}$ and the $\mathrm{pOH}$ does not equal $\mathrm{p}K_\mathrm{b1}$. As well, the two provided dissociation constants for carbonate ion, in the OP's upper figure, are also wrong.

The figure below shows two titration curves: 1) for $\pu{10 mL}$ of $\pu{0.1 M}$ "carbonic acid", with $\pu{0.1 M}$ $\ce{NaOH}$ titrant (black rising curve) and 2) for $\pu{10 mL}$ of $\pu{0.1 M}$ sodium carbonate solution, with $\pu{0.1 M}$ $\ce{HCl}$ titrant (red falling curve):

The $\mathrm{pH}$ of the starting $\pu{0.1 M}$ carbonic acid is 3.67 and the $\mathrm{pH}$ of the starting $\pu{0.1 M}$ sodium carbonate is 11.65. At the first equivalence point, for both titration curves, the $\mathrm{pH}$ is approximately 8.34. The $\mathrm{p}K_\mathrm{a}$ values are as shown in the figure and are from (or derived from) the reference at the bottom.

The next two figures show Excel spreadsheet $\mathrm{pH}$ calculations. The first one shows that $\mathrm{pH} = 8.338$ at the first equivalence point, for the neutralization of sodium carbonate solution with $\ce{HCl}$:

At the first equivalence point, the solution is $\pu{0.05 M}$ sodium bicarbonate plus $\pu{0.05 M}$ $\ce{NaCl}$. Ignoring ionic strength effects, the $\ce{NaCl}$ does nothing to the $\mathrm{pH}$.

The next one shows that $\mathrm{pH} = 8.339$ at the first equivalence point, for the neutralization of carbonic acid with $\ce{NaOH}$:

At the first equivalence point, the solution is $\pu{0.05 M}$ sodium bicarbonate. In both cases, the $\mathrm{pH}$ is approximately 8.34 and is consistent with this answer.

The two $\mathrm{p}K_\mathrm{a}$ values for carbonic acid are from D.G. Harris, In Quantitative Chemical Analysis, 7th Edition; Appendix G; W. H. Freeman and Company, ©2007. The two $\mathrm{p}K_\mathrm{b}$ values for carbonate ion are obtained from the two $\mathrm{p}K_\mathrm{a}$ values for carbonic acid via $\left(\mathrm{p}K_\mathrm{b1} = 14.00 - \mathrm{p}K_\mathrm{a2}\right)$ and $\left(\mathrm{p}K_\mathrm{b2} = 14.00 - \mathrm{p}K_\mathrm{a1}\right)$.

The problem is that both titration curves are wrong: at the first equivalence points, the pH does not equal the relevant pK value. The figure below shows two titration curves: 1) for 0.1 M "carbonic acid", with 0.1 M NaOH titrant (black rising curve) and 2) 0.1 M sodium carbonate solution, with 0.1 M HCl titrant (red falling curve):

At the first equivalence point, for both titration curves, the pH is approximately 8.34. The pK values are as shown in the figure and are from (or derived from) the reference at the bottom.

Just to be as clear as I can be, these next two figures show Excel spreadsheet pH calculations. The first one shows the pH at the first equivalence point, for the neutralization of sodium carbonate solution with HCl:

At the first equivalence point, the solution is 0.05 M sodium bicarbonate plus 0.05 M NaCl. Ignoring ionic strength effects, the NaCl does nothing to the pH.

The next one shows the pH at the first equivalence point, for the neutralization of carbonic acid with NaOH:

At the first equivalence point, the solution is 0.05 M sodium bicarbonate.

All pK values are from (or derived from) D.G. Harris, Quantitative Chemical Analysis, 7th Ed., Appendix G, W.H. Freeman and Company, ©2007.

The problem is that both titration curves are wrong: at the first equivalence points, the pH does not equal $\mathrm{p}K_\mathrm{a1}$ and the pOH does not equal $\mathrm{p}K_\mathrm{b1}$. As well, the two provided dissociation constants for carbonate ion, in the OP's upper figure, are also wrong. 

The figure below shows two titration curves: 1) for 10 mL of 0.1 M "carbonic acid", with 0.1 M NaOH titrant (black rising curve) and 2) for 10 mL of 0.1 M sodium carbonate solution, with 0.1 M HCl titrant (red falling curve):

The pH of the starting 0.1 M carbonic acid is 3.67 and the pH of the starting 0.1 M sodium carbonate is 11.65. At the first equivalence point, for both titration curves, the pH is approximately 8.34. The $\mathrm{p}K$ values are as shown in the figure and are from (or derived from) the reference at the bottom.

The next two figures show Excel spreadsheet pH calculations. The first one shows that pH = 8.338 at the first equivalence point, for the neutralization of sodium carbonate solution with HCl:

At the first equivalence point, the solution is 0.05 M sodium bicarbonate plus 0.05 M NaCl. Ignoring ionic strength effects, the NaCl does nothing to the pH.

The next one shows that pH = 8.339 at the first equivalence point, for the neutralization of carbonic acid with NaOH:

At the first equivalence point, the solution is 0.05 M sodium bicarbonate. In both cases, the pH is approximately 8.34 and is consistent with this answer.

The two $\mathrm{p}K_\mathrm{a}$ values for carbonic acid are from D.G. Harris, Quantitative Chemical Analysis, 7th Ed., Appendix G, W.H. Freeman and Company, ©2007. The two $\mathrm{p}K_\mathrm{b}$ values for carbonate ion are obtained from the two $\mathrm{p}K_\mathrm{a}$ values for carbonic acid via $\mathrm{p}K_\mathrm{b1}$ = 14.00 - $\mathrm{p}K_\mathrm{a2}$ and $\mathrm{p}K_\mathrm{b2}$ = 14.00 - $\mathrm{p}K_\mathrm{a1}$.

The problem is that both titration curves are wrong: at the first equivalence points, the pH does not equal the relevant pK value. The titration of 0.1 M "carbonic acid", with 0.1 M NaOH, is this:

At the first equivalence point, after 10 mL of 0.1 M NaOH has been added, the pH is approximately 8.34. Half-way to the first equivalence point, i.e., after 5 mL of 0.1 M NaOH has been added, the pH equals 6.35, as expected. The pK values I used are 6.351 and 10.329, from the reference at the bottom.

Just to be as clear as I can be, these next two figures show Excel spreadsheet pH calculations. The first one shows the pH at the first equivalence point, for the neutralization of sodium carbonate solution with HCl:

At the first equivalence point, the solution is 0.05 M sodium bicarbonate plus 0.05 M NaCl. Ignoring ionic strength effects, the NaCl does nothing to the pH.

The next one shows the pH at the first equivalence point, for the neutralization of carbonic acid with NaOH:

At the first equivalence point, the solution is 0.05 M sodium bicarbonate.

All pK values are from (or derived from) D.G. Harris, Quantitative Chemical Analysis, 7th Ed., Appendix G, W.H. Freeman and Company, ©2007.

The problem is that both titration curves are wrong: at the first equivalence points, the pH does not equal the relevant pK value. The figure below shows two titration curves: 1) for 0.1 M "carbonic acid", with 0.1 M NaOH titrant (black rising curve) and 2) 0.1 M sodium carbonate solution, with 0.1 M HCl titrant (red falling curve):

At the first equivalence point, for both titration curves, the pH is approximately 8.34. The pK values are as shown in the figure and are from (or derived from) the reference at the bottom.

Just to be as clear as I can be, these next two figures show Excel spreadsheet pH calculations. The first one shows the pH at the first equivalence point, for the neutralization of sodium carbonate solution with HCl:

At the first equivalence point, the solution is 0.05 M sodium bicarbonate plus 0.05 M NaCl. Ignoring ionic strength effects, the NaCl does nothing to the pH.

The next one shows the pH at the first equivalence point, for the neutralization of carbonic acid with NaOH:

At the first equivalence point, the solution is 0.05 M sodium bicarbonate.

All pK values are from (or derived from) D.G. Harris, Quantitative Chemical Analysis, 7th Ed., Appendix G, W.H. Freeman and Company, ©2007.