Lawrence, 29, was recalled after an international absence of almost two years, having averaged 26.84 across 14 previous Tests including a stint as a makeshift opener.
By picking him, England prioritised having four pace bowling options over a frontline spinner, with Lawrence backed to provided the overs of slow bowling if needed. He has only bowled one over in the series so far with pace bowlers dominant.
On Saturday Lawrence made his mark by coming to the crease at 83-4 and steering England to 177-7 in their second innings when rain brought an early close.
The right-hander, whose best Test score remains 91, countered excessive seam movement and uneven bounce by showing attacking intent, hitting six fours and two sixes.
“He seems to read the game,” Vaughan said. “It was the right way to play at that moment.
“He has a quirky technique that has worked for him and is very confident. He knows his scoring areas.
“He is clearly a better player than when he played for England last time.”
England’s current balance – with Lawrence picked over a frontline spinner – appears suited to home conditions, and could also work in South Africa and Australia where England play this winter.
While stronger spin options would likely be needed for the tour of Bangladesh in the new year, the success of England’s pace quartet of Ollie Robinson, Jofra Archer, Gus Atkinson and Josh Tongue has lessened the need for a spinner in home conditions and, in turn, helped Lawrence’s cause.
Key selection decisions are still looming, however, with Jacob Bethell currently out of the series because of a knee injury.
Bethell, who also bowls part-time spin, could replace Lawrence, or regain his place from Jordan Cox, who has impressed with two fifties while filling in for Bethell at number three.
Other options include pushing Cox or Bethell up to open in place of Emilio Gay – Cox opened early in his career in county cricket and Bethell is England’s current 50-over opener – and moving Cox down the order to replace Jamie Smith as wicketkeeper.
Smith scored an impressive 61 in England’s first innings and has improved his glovework after a poor showing in the Ashes.
“It’s pointless looking to the future because it hasn’t happened yet,” Lawrence said. “I’m just trying to contribute every Test I can and give a good showing of myself.
“Thankfully, so far, I think I have done.
“Obviously thoughts creep into your head and I’m desperate, along with all the other boys, to establish myself in an England shirt, which is something I haven’t quite done in my career so far.
“It’s just about trying to take every opportunity you get and really enjoy the occasion instead of feeling nervous about the occasion. Just have fun, play the cricket that’s got me here this summer.”