Graph labelling where vertices with a common neighbour get different labels
`Special labelling' is not exactly $L(0,1)$-coloring, but is very close. In $L(0,1)$-coloring, neighboring vertices can get the same colour even if they have a common neighbor. Speciall labelling do not allow this. Special labelling is already studied in the literature under the name injective coloring.

An injective colouring of a graph $G$ is a colouring $c$ of the vertices
of $G$ that assigns different colours to any pair of vertices that have a
common neighbour.

There are a number of papers on injective coloring. The main graph classes studied are planar graphs [1,2,3], chordal graphs [4], regular graphs [5], and sparse graphs [6]
References
[1] Bu, Yuehua; Qi, Chentao; Zhu, Junlei; Xu, Ting, Injective coloring of planar graphs,  ZBL07300884.
[2] Lužar, Borut; Škrekovski, Riste; Tancer, Martin, Injective colorings of planar graphs with few colors, Discrete Math. 309, No. 18, 5636-5649 (2009). ZBL1209.05093.
[3] Bu, Yuehua; Wang, Chao; Yang, Sheng, List injective coloring of planar graphs.,  ZBL06986943.
[4] Hell, Pavol; Raspaud, André; Stacho, Juraj, On injective colourings of chordal graphs, Laber, Eduardo Sany (ed.) et al., LATIN 2008 Berlin: Springer (ISBN 978-3-540-78772-3/pbk). LNCS 4957, 520-530 (2008). ZBL1136.68463.
[5] Hahn, Geňa; Kratochvíl, Jan; Širáň, Jozef; Sotteau, Dominique, On the injective chromatic number of graphs, Discrete Math. 256, No. 1-2, 179-192 (2002). ZBL1007.05046.
[6] Cranston, Daniel W.; Kim, Seog-Jin; Yu, Gexin, Injective colorings of sparse graphs, Discrete Math. 310, No. 21, 2965-2973 (2010). ZBL1209.05075.