I don't know that my reply was helpful - we've yet to discover the maker
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your description of the top of your vase suggests the glass was hot finished at the rim - (not cold finished, which leaves a flat sharp-edged rim) - and this would look to agree with the fact that the vase was attached, at the base, to a pontil rod, in order to heat finish the top rim.
When the rod is snapped away from the base, the result is a rough/sharp area which is generally described as a pontil scar, and this scar needs to be removed by grinding/polishing, the consequence of which leaves the usual tell-tale shallow depression on the underside of the base.
There has been a mould involved in the making of your vase, though quite what sort and to what extent I'm unsure - there are other folk here who will hopefully explain that part of the process for you.
It is quite remarkable how similar in appearance these surface 'ovals/lenses' are to the B. S. Large Olives pattern, but from what you've said, it seems that yours is unrelated.
So - no further forward unfortunately, but hopefully someone will be able to help.
P.S. It's worth remembering that genuine ground/polished pontil depressions - on the underside of glass - can be mimicked by moulding. Some modern inexpensive glass pieces are known to show what appears to be a polished depression, but looked at carefully it can be seen that the depression has been mould-formed, and not made in the traditional manner.
Not suggesting your piece is so formed, but just a snippet of information that might help the inexperienced.