I feel like shouting: THOSE ARE GRAPE VINES.
Now, I feel better.
I would like to offer some advice to new gardeners who are learning ID characteristics, and trying to sort out what can be confusing terminology and irritating similarities. Do not go to an obscure and often difficult to determine feature (pith color) for the definitive ID, unless/until you have eliminated all of the regular/normal ID features. As you have observed, the pith color on an older (more than one year) stem can be different in color than the pith of new growth.
Some of these usual ID characteristics include: Orientation of leaves/buds/stems along the main trunk or branches; leaf morphology; flowering characteristics; fruiting characteristics; with vines, the characteristics of how they climb or attach themselves; etc.
It would be more helpful to have more pictures of (more than one) leaves, post them here, and request review. Photograph how the leaves are attached along the stems, including closeups of the petioles, the bud scars, the leaf scars, the tendrils (!), and any varying bark characteristics along those stems.
Vitis sp. are markedly different from Ampelopsis sp., and not just by pith color. I think you are right to keep that vine for the rest of the season, and observe all the characteristics it displays in that time. I think it will also be worthwhile to try to find known examples of Ampelopsis sp. and Vitis sp., so you can compare/contrast them in real time.
One of the great things about having a horticultural background AND working in a municipal parks department, is/was the opportunity to see all these plants all the time. Upon reading a post like this, and believing to myself "well, those two species are nothing alike", I could go out and find samples of both. I could review to myself WHY I believe they are nothing alike, and try to look at them with fresh eyes to see where those actual differences are. Then, taking pictures of the definitive characteristics, I could post them to explain those things.
I would sincerely ask you to reconsider your position. Take a very close look at the vein pattern in the leaves on your plant, and then the vein pattern on the leaves in the Ampelopsis links supplied by plantladylin. I believe they are markedly different, but I'm lacking the botanical terminology memory to say what each pattern is called.
With distancing going on, I'm not out on the ground with all those plants in the parks. I have a wealth of volunteer Vitis at Viburnum Valley, but thankfully have zero of the pernicious pest Ampelopsis brevipedunculata.
So, we rest and await your patience with us. Please provide some more evidence, and help us assure you that it is NOT unlikely at all for you to have a volunteer grape vine. It likely originated exactly like the wild blackberries - from free contributions from bird depositions.
I hope your bare root grapes get off to a good start. Comparison of those plants to your volunteer may also "bear fruit". In fact, that made me think of another separating ID feature - the root system characteristics.