Despite the recent availability of large samples of stars with high-precision Li abundances, there are many unanswered questions about the evolution of this unique element in the Galaxy and in the stars themselves. It is unclear which parameters and physical mechanisms govern Li depletion in late-type stars and if Galactic enrichment has proceeded differently in different stellar populations. With this study we aim to explore these questions further by mapping the evolution of Li with stellar mass, age, and effective temperature for Milky Way disk stars, linking the metal-poor and metal-rich regimes, and how Li differs in the thin and thick disks. We determine Li abundances for a well-studied sample of 714 F and G dwarf, turn-off, and subgiant stars in the solar neighbourhood. The analysis is based on line synthesis of the 7Li line at 6707 Å in high-resolution and high-signal-to-noise ratio echelle spectra, obtained with the MIKE, FEROS, SOFIN, UVES, and FIES spectrographs. The presented Li abundances are corrected for non-LTE effects. Out of the sample of 714 stars, we are able to determine Li abundances for 394 stars and upper limits on the Li abundance for another 121 stars. Out of 36 stars that are listed as exoplanet host stars, 18 have well-determined Li abundances and 6 have Li upper limits. Our main finding is that there are no signatures of Li production in stars associated with the thick disk. Instead the Li abundance trend is decreasing with metallicity for these thick disk stars. Significant Li production is however seen in the thin disk, with a steady increase towards super-solar metallicities. At the highest metallicities, however, around [Fe/H] ≈ +0.3, we tentatively confirm the recent discovery that the Li abundances level out. Our finding contradicts the other recent studies that found that Li is also produced in the thick disk. We find that this is likely due to the α-enhancement criteria which those studies used to define their thick disk samples. By using the more robust age criteria, we are able to define a thick disk stellar sample that is much less contaminated by thin disk stars. Furthermore, we also tentatively confirm the age-Li correlation for solar twin stars, and we find that there is no correlation between Li abundance and whether the stars have detected exoplanets or not. The major conclusion that can be drawn from this study is that no significant Li production relative to the primordial abundance took place during the first few billion years of the Milky Way, an era coinciding with the formation and evolution of the thick disk. Significant Li enrichment then took place once long-lived low-mass stars (acting on a timescale longer than SNIa) had had time to contribute to the chemical enrichment of the interstellar medium.